import Mathlib.Analysis.Analytic.Polynomial import Mathlib.Analysis.Calculus.Deriv.Polynomial import Mathlib.Analysis.Calculus.ContDiff.RCLike import Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas import Mathlib.Analysis.Calculus.Taylor import Mathlib.Analysis.Complex.LocallyUniformLimit import Mathlib.Analysis.ODE.ExistUnique import Mathlib.Analysis.SpecialFunctions.Complex.Analytic import Mathlib.Analysis.SpecialFunctions.Pow.Deriv import Mathlib.Analysis.SpecialFunctions.Sqrt import Mathlib.RingTheory.PowerSeries.Derivative import Mathlib.Topology.ContinuousMap.Compact import Mathlib.Tactic.ComputeDegree import Mathlib.Tactic.FieldSimp import Mathlib.Tactic.FunProp import Mathlib.Tactic.Linarith import Mathlib.Tactic.LinearCombination import Mathlib.Tactic.NormNum import Mathlib.Tactic.Positivity import Mathlib.Tactic.Push import Mathlib.Tactic.Ring namespace OAI section section SourceConstruction /-! The normalized sonic equation is regular singular. After removing its nonresonant finite jet, the remaining problem has the form `z g' + c g = z H(z, g)` with `c > 0`. The analytic disk space below supports a contraction argument for this equation. -/ noncomputable section SourceNoncomp open Set Filter Metric MeasureTheory open scoped Topology NNReal namespace SepticProfile.Fuchsian abbrev Disk := ↥(closedBall (0 : ℂ) 1) abbrev DiskMap := C(Disk, ℂ) def extend (f : DiskMap) (z : ℂ) : ℂ := by classical exact if h : z ∈ closedBall (0 : ℂ) 1 then f ⟨z,h⟩ else 0 @[simp] theorem extend_coe (f : DiskMap) (z : Disk) : extend f z = f z := by simp [extend, z.property] @[simp] theorem extend_zero : extend (0 : DiskMap) = 0 := by funext z simp [extend] @[simp] theorem extend_add (f g : DiskMap) : extend (f+g) = extend f + extend g := by funext z simp only [extend, Pi.add_apply] split <;> simp @[simp] theorem extend_smul (c : ℂ) (f : DiskMap) : extend (c • f) = c • extend f := by funext z simp only [extend, Pi.smul_apply] split <;> simp def analyticDisk : Submodule ℂ DiskMap where carrier := {f | DifferentiableOn ℂ (extend f) (ball (0 : ℂ) 1)} zero_mem' := by simp add_mem' := by intro f g hf hg simpa using hf.add hg smul_mem' := by intro c f hf simpa using hf.const_smul c theorem analyticDisk_closed : IsClosed (analyticDisk : Set DiskMap) := by apply IsSeqClosed.isClosed intro fs f hfs hf change DifferentiableOn ℂ (extend f) (ball (0 : ℂ) 1) have hu : TendstoUniformlyOn (fun n => extend (fs n)) (extend f) atTop (ball (0 : ℂ) 1) := by rw [Metric.tendstoUniformlyOn_iff] intro ε hε have he : ∀ᶠ n in atTop, dist (fs n) f < ε := hf.eventually (ball_mem_nhds f hε) filter_upwards [he] with n hn z hz have hz' : z ∈ closedBall (0 : ℂ) 1 := ball_subset_closedBall hz have hb := ContinuousMap.dist_apply_le_dist (f := fs n) (g := f) ⟨z,hz'⟩ simpa [extend, hz', dist_comm] using hb.trans_lt hn exact hu.tendstoLocallyUniformlyOn.differentiableOn (Eventually.of_forall hfs) isOpen_ball instance : CompleteSpace analyticDisk := analyticDisk_closed.completeSpace_coe abbrev UnitInterval := ↥(Icc (0 : ℝ) 1) def diskScale (t : UnitInterval) (z : Disk) : Disk := ⟨(t : ℝ) • (z : ℂ), by rw [mem_closedBall_zero_iff, norm_smul, Real.norm_eq_abs, abs_of_nonneg t.property.1] exact (mul_le_mul_of_nonneg_left (mem_closedBall_zero_iff.mp z.property) t.property.1).trans (by simpa using t.property.2)⟩ theorem continuous_diskScale : Continuous (fun p : UnitInterval × Disk => diskScale p.1 p.2) := by apply Continuous.subtype_mk exact (continuous_subtype_val.comp continuous_fst).smul (continuous_subtype_val.comp continuous_snd) theorem scale_mem_ball (t : UnitInterval) {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) : (t : ℝ) • z ∈ ball (0 : ℂ) 1 := by rw [mem_ball_zero_iff] at hz ⊢ rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg t.property.1] exact (mul_le_mul_of_nonneg_right t.property.2 (norm_nonneg z)).trans_lt (by simpa using hz) def dilateMap (t : UnitInterval) (f : DiskMap) : DiskMap := ⟨fun z => f (diskScale t z), f.continuous.comp (continuous_diskScale.comp (continuous_const.prodMk continuous_id))⟩ theorem extend_dilate_on (t : UnitInterval) (f : DiskMap) {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) : extend (dilateMap t f) z = extend f ((t : ℝ) • z) := by have hz' := ball_subset_closedBall hz have htz := ball_subset_closedBall (scale_mem_ball t hz) unfold extend rw [dite_eq_left hz', dite_eq_left htz] rfl theorem dilate_analytic (t : UnitInterval) {f : DiskMap} (hf : f ∈ analyticDisk) : dilateMap t f ∈ analyticDisk := by change DifferentiableOn ℂ (extend (dilateMap t f)) (ball (0 : ℂ) 1) change DifferentiableOn ℂ (extend f) (ball (0 : ℂ) 1) at hf have hs : DifferentiableOn ℂ (fun z : ℂ => (t : ℝ) • z) (ball 0 1) := by simpa only [Complex.real_smul, id_eq] using (differentiableOn_id.const_mul (t : ℂ)) exact (hf.comp hs (fun z hz => scale_mem_ball t hz)).congr (fun z hz => extend_dilate_on t f hz) def dilate (t : UnitInterval) (f : analyticDisk) : analyticDisk := ⟨dilateMap t f, dilate_analytic t f.property⟩ theorem norm_dilate_le (t : UnitInterval) (f : analyticDisk) : ‖dilate t f‖ ≤ ‖f‖ := by change ‖dilateMap t f‖ ≤ ‖(f : DiskMap)‖ apply (ContinuousMap.norm_le _ (norm_nonneg _)).2 intro z exact (f : DiskMap).norm_coe_le_norm (diskScale t z) theorem continuous_dilate (f : analyticDisk) : Continuous (fun t => dilate t f) := by apply Continuous.subtype_mk apply ContinuousMap.continuous_of_continuous_uncurry exact (f : DiskMap).continuous.comp continuous_diskScale def clip (t : ℝ) : UnitInterval := projIcc 0 1 (by norm_num) t def radialIntegrand (c : ℝ) (f : analyticDisk) (t : ℝ) : analyticDisk := (t ^ c : ℝ) • dilate (clip t) f def radialIntegral (c : ℝ) (f : analyticDisk) : analyticDisk := ∫ t : ℝ in 0..1, radialIntegrand c f t theorem continuous_radialIntegrand {c : ℝ} (hc : 0 ≤ c) (f : analyticDisk) : Continuous (radialIntegrand c f) := by exact (Real.continuous_rpow_const hc).smul ((continuous_dilate f).comp (continuous_projIcc (h := show (0 : ℝ) ≤ 1 by norm_num))) theorem norm_radialIntegrand_le {c t : ℝ} (hc : 0 ≤ c) (ht : t ∈ Icc (0 : ℝ) 1) (f : analyticDisk) : ‖radialIntegrand c f t‖ ≤ ‖f‖ := by rw [radialIntegrand, norm_smul, Real.norm_eq_abs, abs_of_nonneg (Real.rpow_nonneg ht.1 _)] calc t ^ c * ‖dilate (clip t) f‖ ≤ 1 * ‖f‖ := mul_le_mul (Real.rpow_le_one ht.1 ht.2 hc) (norm_dilate_le (clip t) f) (norm_nonneg (dilate (clip t) f)) (by norm_num : (0 : ℝ) ≤ 1) _ = ‖f‖ := one_mul _ theorem norm_radialIntegral_le {c : ℝ} (hc : 0 ≤ c) (f : analyticDisk) : ‖radialIntegral c f‖ ≤ ‖f‖ := by change ‖∫ t : ℝ in 0..1, radialIntegrand c f t‖ ≤ ‖f‖ have hb : ∀ t ∈ uIoc (0 : ℝ) 1, ‖radialIntegrand c f t‖ ≤ ‖f‖ := by intro t ht have ht' : t ∈ Ioc (0 : ℝ) 1 := by simpa only [uIoc_of_le (by norm_num : (0:ℝ) ≤ 1)] using ht exact norm_radialIntegrand_le hc ⟨ht'.1.le, ht'.2⟩ f calc ‖∫ t : ℝ in 0..1, radialIntegrand c f t‖ ≤ ‖f‖ * |(1 : ℝ) - 0| := intervalIntegral.norm_integral_le_of_norm_le_const hb _ = ‖f‖ := by rw [sub_zero, abs_one, mul_one] theorem differentiableAt_extend (f : analyticDisk) {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) : DifferentiableAt ℂ (extend (f : DiskMap)) z := f.property.differentiableAt (isOpen_ball.mem_nhds hz) theorem deriv_bound (f : analyticDisk) {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) : ‖deriv (extend (f : DiskMap)) z‖ ≤ ‖f‖ / ((1 - ‖z‖) / 2) := by have hz' : ‖z‖ < 1 := mem_ball_zero_iff.mp hz have hr : 0 < (1 - ‖z‖) / 2 := by linarith have hs : closedBall z ((1 - ‖z‖) / 2) ⊆ ball (0 : ℂ) 1 := by intro w hw rw [mem_ball_zero_iff] have hw' : ‖w-z‖ ≤ (1-‖z‖)/2 := mem_closedBall_iff_norm.mp hw have hb : ‖w‖ ≤ ‖w-z‖ + ‖z‖ := by simpa only [sub_add_cancel] using norm_add_le (w-z) z linarith rw [← Complex.cderiv_eq_deriv isOpen_ball f.property hr hs] apply Complex.norm_cderiv_le hr intro w hw have hw' : w ∈ closedBall (0 : ℂ) 1 := ball_subset_closedBall (hs (sphere_subset_closedBall hw)) have hn := (f : DiskMap).norm_coe_le_norm ⟨w,hw'⟩ rw [← extend_coe] at hn exact hn def derivLinear {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) : analyticDisk →ₗ[ℂ] ℂ where toFun f := deriv (extend (f : DiskMap)) z map_add' f g := by simp only [Submodule.coe_add, extend_add] exact deriv_add (differentiableAt_extend f hz) (differentiableAt_extend g hz) map_smul' c f := by simp only [Submodule.coe_smul_of_tower, extend_smul, RingHom.id_apply] exact deriv_const_smul c (differentiableAt_extend f hz) def derivEval {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) : analyticDisk →L[ℂ] ℂ := (derivLinear hz).mkContinuous (((1-‖z‖)/2)⁻¹) (by intro f simpa only [derivLinear, LinearMap.coe_mk, AddHom.coe_mk, div_eq_mul_inv, mul_comm] using deriv_bound f hz) def eval (z : Disk) : analyticDisk →L[ℂ] ℂ := (ContinuousMap.evalCLM ℂ z).comp analyticDisk.subtypeL @[simp] theorem eval_apply (z : Disk) (f : analyticDisk) : eval z f = (f : DiskMap) z := rfl theorem deriv_dilate (t : UnitInterval) (f : analyticDisk) {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) : deriv (extend (dilate t f : DiskMap)) z = (t : ℂ) * deriv (extend (f : DiskMap)) ((t : ℂ) * z) := by have ht := scale_mem_ball t hz rw [Complex.real_smul] at ht have hd := (differentiableAt_extend f ht).hasDerivAt.comp z ((hasDerivAt_id z).const_mul (t : ℂ)) have he : (fun w => extend (f : DiskMap) ((t : ℂ) * w)) =ᶠ[𝓝 z] extend (dilate t f : DiskMap) := by filter_upwards [isOpen_ball.mem_nhds hz] with w hw change extend (f : DiskMap) ((t : ℂ) * w) = extend (dilateMap t (f : DiskMap)) w simpa only [Complex.real_smul] using (extend_dilate_on t (f : DiskMap) hw).symm simpa only [mul_one, mul_comm] using (hd.congr_of_eventuallyEq he.symm).deriv theorem clip_coe {t : ℝ} (ht : t ∈ Icc (0 : ℝ) 1) : (clip t : ℝ) = t := congrArg Subtype.val (projIcc_of_mem (by norm_num) ht) theorem radialIntegrand_eval {c t : ℝ} (f : analyticDisk) (ht : t ∈ Icc (0 : ℝ) 1) {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) : eval ⟨z,ball_subset_closedBall hz⟩ (radialIntegrand c f t) = ((t ^ c : ℝ) : ℂ) * extend (f : DiskMap) ((t : ℂ) * z) := by change (t ^ c : ℝ) • (f : DiskMap) (diskScale (clip t) ⟨z,ball_subset_closedBall hz⟩) = _ rw [Complex.real_smul, ← extend_coe] congr 2 change (clip t : ℝ) • z = (t : ℂ) * z rw [clip_coe ht, Complex.real_smul] theorem radialIntegrand_deriv {c t : ℝ} (f : analyticDisk) (ht : t ∈ Icc (0 : ℝ) 1) {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) : derivEval hz (radialIntegrand c f t) = ((t ^ c : ℝ) : ℂ) * (t : ℂ) * deriv (extend (f : DiskMap)) ((t : ℂ) * z) := by change deriv (extend ((radialIntegrand c f t : analyticDisk) : DiskMap)) z = _ have he : extend ((radialIntegrand c f t : analyticDisk) : DiskMap) = ((t ^ c : ℝ) : ℂ) • extend (dilate (clip t) f : DiskMap) := by funext w classical by_cases hw : w ∈ closedBall (0 : ℂ) 1 · simp only [extend, dite_eq_left hw, Pi.smul_apply] change (t ^ c : ℝ) • (dilate (clip t) f : DiskMap) ⟨w,hw⟩ = ((t ^ c : ℝ) : ℂ) • (dilate (clip t) f : DiskMap) ⟨w,hw⟩ rw [Complex.real_smul, smul_eq_mul] · simp only [extend, dite_eq_right hw, Pi.smul_apply, smul_zero] rw [he, deriv_const_smul _ (differentiableAt_extend (dilate (clip t) f) hz), deriv_dilate (clip t) f hz] rw [clip_coe ht] simp only [smul_eq_mul, mul_assoc] def radialOperator (c : ℝ) {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) : analyticDisk →L[ℂ] ℂ := z • derivEval hz + ((c + 1 : ℝ) : ℂ) • eval ⟨z,ball_subset_closedBall hz⟩ theorem radialOperator_apply (c : ℝ) {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) (f : analyticDisk) : radialOperator c hz f = z * deriv (extend (f : DiskMap)) z + ((c+1 : ℝ) : ℂ) * extend (f : DiskMap) z := by change z * deriv (extend (f : DiskMap)) z + ((c+1 : ℝ) : ℂ) * (f : DiskMap) ⟨z,ball_subset_closedBall hz⟩ = _ rw [← extend_coe] theorem radial_primitive_deriv {c : ℝ} (f : analyticDisk) {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) {t : ℝ} (ht : t ∈ Ioo (0 : ℝ) 1) : HasDerivAt (fun x : ℝ => (x ^ (c+1) : ℝ) • extend (f : DiskMap) ((x : ℂ) * z)) (radialOperator c hz (radialIntegrand c f t)) t := by have htz : (t : ℂ) * z ∈ ball (0 : ℂ) 1 := by simpa only [Complex.real_smul] using scale_mem_ball ⟨t,ht.1.le,ht.2.le⟩ hz have hf := ((differentiableAt_extend f htz).hasDerivAt.comp (t : ℂ) ((hasDerivAt_id (t : ℂ)).mul_const z)).comp_ofReal have hp := Real.hasDerivAt_rpow_const (p := c+1) (Or.inl (ne_of_gt ht.1)) have h := hp.smul hf have he : radialOperator c hz (radialIntegrand c f t) = (t ^ (c+1) : ℝ) • (deriv (extend (f : DiskMap)) ((t : ℂ) * z) * (1 * z)) + ((c+1) * t ^ (c+1-1) : ℝ) • extend (f : DiskMap) ((t : ℂ) * z) := by rw [radialOperator, add_apply, smul_apply, smul_apply, radialIntegrand_eval f ⟨ht.1.le,ht.2.le⟩ hz, radialIntegrand_deriv f ⟨ht.1.le,ht.2.le⟩ hz] simp only [Complex.real_smul, smul_eq_mul, Complex.ofReal_mul, add_sub_cancel_right, one_mul] rw [Real.rpow_add_one ht.1.ne'] push_cast ring rw [he] exact h theorem radial_primitive_continuous {c : ℝ} (hc : 0 ≤ c) (f : analyticDisk) {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) : ContinuousOn (fun x : ℝ => (x ^ (c+1) : ℝ) • extend (f : DiskMap) ((x : ℂ) * z)) (Icc (0 : ℝ) 1) := by have hC : Continuous (fun x : ℝ => (x ^ (c+1) : ℝ) • eval ⟨z,ball_subset_closedBall hz⟩ (dilate (clip x) f)) := (Real.continuous_rpow_const (by linarith : 0 ≤ c+1)).smul ((eval ⟨z,ball_subset_closedBall hz⟩).continuous.comp ((continuous_dilate f).comp (continuous_projIcc (h := show (0 : ℝ) ≤ 1 by norm_num)))) apply hC.continuousOn.congr intro t ht apply congrArg (fun w : ℂ => (t ^ (c+1) : ℝ) • w) change extend (f : DiskMap) ((t : ℂ) * z) = (f : DiskMap) (diskScale (clip t) ⟨z,ball_subset_closedBall hz⟩) rw [← extend_coe] congr 1 change (t : ℂ) * z = (clip t : ℝ) • z rw [clip_coe ht, Complex.real_smul] theorem radialIntegral_ode {c : ℝ} (hc : 0 ≤ c) (f : analyticDisk) {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) : radialOperator c hz (radialIntegral c f) = extend (f : DiskMap) z := by have hi : IntervalIntegrable (radialIntegrand c f) volume 0 1 := (continuous_radialIntegrand hc f).intervalIntegrable _ _ have hL : IntervalIntegrable (fun t => radialOperator c hz (radialIntegrand c f t)) volume 0 1 := ((radialOperator c hz).continuous.comp (continuous_radialIntegrand hc f)).intervalIntegrable _ _ have hFTC := intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le (by norm_num : (0 : ℝ) ≤ 1) (radial_primitive_continuous hc f hz) (fun t ht => radial_primitive_deriv f hz ht) hL rw [(radialOperator c hz).intervalIntegral_comp_comm hi] at hFTC change radialOperator c hz (radialIntegral c f) = _ at hFTC simpa only [Real.one_rpow, Real.zero_rpow (by linarith : c+1 ≠ 0), one_smul, zero_smul, sub_zero, Complex.ofReal_one, Complex.ofReal_zero, one_mul, zero_mul] using hFTC def coordinateMulMap (f : DiskMap) : DiskMap := ⟨fun z => (z : ℂ) * f z, continuous_subtype_val.mul f.continuous⟩ theorem extend_coordinateMulMap (f : DiskMap) : extend (coordinateMulMap f) = fun z => z * extend f z := by funext z classical by_cases hz : z ∈ closedBall (0 : ℂ) 1 · simp only [extend, dite_eq_left hz] rfl · simp only [extend, dite_eq_right hz, mul_zero] def coordinateMul (f : analyticDisk) : analyticDisk := ⟨coordinateMulMap (f : DiskMap), by change DifferentiableOn ℂ (extend (coordinateMulMap (f : DiskMap))) (ball 0 1) rw [extend_coordinateMulMap] exact differentiableOn_id.mul f.property⟩ theorem norm_coordinateMul_le (f : analyticDisk) : ‖coordinateMul f‖ ≤ ‖f‖ := by change ‖coordinateMulMap (f : DiskMap)‖ ≤ ‖(f : DiskMap)‖ apply (ContinuousMap.norm_le _ (norm_nonneg _)).2 intro z change ‖(z : ℂ) * (f : DiskMap) z‖ ≤ ‖(f : DiskMap)‖ rw [norm_mul] calc ‖(z : ℂ)‖ * ‖(f : DiskMap) z‖ ≤ 1 * ‖(f : DiskMap)‖ := mul_le_mul (mem_closedBall_zero_iff.mp z.property) ((f : DiskMap).norm_coe_le_norm z) (norm_nonneg _) (by norm_num) _ = ‖(f : DiskMap)‖ := one_mul _ theorem coordinateMul_sub (f g : analyticDisk) : coordinateMul (f-g) = coordinateMul f - coordinateMul g := by apply Subtype.ext apply ContinuousMap.ext intro z exact mul_sub (z : ℂ) ((f : DiskMap) z) ((g : DiskMap) z) theorem radialIntegrand_sub (c : ℝ) (f g : analyticDisk) (t : ℝ) : radialIntegrand c (f-g) t = radialIntegrand c f t - radialIntegrand c g t := by apply Subtype.ext apply ContinuousMap.ext intro z exact smul_sub (t ^ c : ℝ) ((f : DiskMap) (diskScale (clip t) z)) ((g : DiskMap) (diskScale (clip t) z)) theorem radialIntegral_sub {c : ℝ} (hc : 0 ≤ c) (f g : analyticDisk) : radialIntegral c (f-g) = radialIntegral c f - radialIntegral c g := by unfold radialIntegral simp_rw [radialIntegrand_sub] exact intervalIntegral.integral_sub (continuous_radialIntegrand hc f |>.intervalIntegrable _ _) (continuous_radialIntegrand hc g |>.intervalIntegrable _ _) def inverse (c : ℝ) (f : analyticDisk) : analyticDisk := coordinateMul (radialIntegral c f) theorem norm_inverse_le {c : ℝ} (hc : 0 ≤ c) (f : analyticDisk) : ‖inverse c f‖ ≤ ‖f‖ := (norm_coordinateMul_le _).trans (norm_radialIntegral_le hc f) theorem inverse_sub {c : ℝ} (hc : 0 ≤ c) (f g : analyticDisk) : inverse c (f-g) = inverse c f - inverse c g := by rw [inverse, radialIntegral_sub hc, coordinateMul_sub] rfl theorem inverse_ode {c : ℝ} (hc : 0 ≤ c) (f : analyticDisk) {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) : z * deriv (extend (inverse c f : DiskMap)) z + (c : ℂ) * extend (inverse c f : DiskMap) z = z * extend (f : DiskMap) z := by have he : extend (inverse c f : DiskMap) = fun z => z * extend (radialIntegral c f : DiskMap) z := extend_coordinateMulMap _ have hd : deriv (fun w => w * extend (radialIntegral c f : DiskMap) w) z = extend (radialIntegral c f : DiskMap) z + z * deriv (extend (radialIntegral c f : DiskMap)) z := by simpa only [id_eq, Pi.mul_def, one_mul] using ((hasDerivAt_id z).mul (differentiableAt_extend (radialIntegral c f) hz).hasDerivAt).deriv rw [he, hd] have h := radialIntegral_ode hc f hz rw [radialOperator_apply] at h rw [← h] push_cast ring theorem inverse_zero (c : ℝ) (f : analyticDisk) : extend (inverse c f : DiskMap) 0 = 0 := by rw [show extend (inverse c f : DiskMap) = (fun z => z * extend (radialIntegral c f : DiskMap) z) from extend_coordinateMulMap _] exact zero_mul _ /-- Bounds on an analytic right-hand side give an invariant fixed point of the regular-singular integral operator. -/ theorem exists_fuchsian_disk_invariant {c r R M L : ℝ} (hc : 0 ≤ c) (hr : 0 ≤ r) (hR : 0 ≤ R) (hrM : r * M ≤ R) (hrL : r * L ≤ 1/2) (H : analyticDisk → analyticDisk) (hbound : ∀ f, ‖f‖ ≤ R → ‖H f‖ ≤ M) (hlip : ∀ f g, ‖f‖ ≤ R → ‖g‖ ≤ R → ‖H f - H g‖ ≤ L * ‖f-g‖) (P : Set analyticDisk) (hPc : IsClosed P) (hP0 : 0 ∈ P) (hP : ∀ f ∈ P, ‖f‖ ≤ R → (r : ℂ) • inverse c (H f) ∈ P) : ∃ f : analyticDisk, f ∈ P ∧ ‖f‖ ≤ R ∧ extend (f : DiskMap) 0 = 0 ∧ ∀ z ∈ ball (0 : ℂ) 1, z * deriv (extend (f : DiskMap)) z + (c : ℂ) * extend (f : DiskMap) z = (r : ℂ) * z * extend (H f : DiskMap) z := by let T : analyticDisk → analyticDisk := fun f => (r : ℂ) • inverse c (H f) have hT : ∀ f, ‖f‖ ≤ R → ‖T f‖ ≤ R := by intro f hf change ‖(r : ℂ) • inverse c (H f)‖ ≤ R rw [norm_smul, Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg hr] exact (mul_le_mul_of_nonneg_left ((norm_inverse_le hc (H f)).trans (hbound f hf)) hr).trans hrM have hTd : ∀ f g, ‖f‖ ≤ R → ‖g‖ ≤ R → ‖T f-T g‖ ≤ (1/2 : ℝ) * ‖f-g‖ := by intro f g hf hg change ‖(r : ℂ) • inverse c (H f) - (r : ℂ) • inverse c (H g)‖ ≤ _ rw [← smul_sub (r : ℂ) (inverse c (H f)) (inverse c (H g)), ← inverse_sub hc, norm_smul, Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg hr] calc r * ‖inverse c (H f-H g)‖ ≤ r * (L * ‖f-g‖) := mul_le_mul_of_nonneg_left ((norm_inverse_le hc (H f-H g)).trans (hlip f g hf hg)) hr _ ≤ (1/2 : ℝ) * ‖f-g‖ := by rw [← mul_assoc] exact mul_le_mul_of_nonneg_right hrL (norm_nonneg _) have hmaps : MapsTo T (closedBall 0 R) (closedBall 0 R) := by intro f hf exact mem_closedBall_zero_iff.mpr (hT f (mem_closedBall_zero_iff.mp hf)) have hmapsP : MapsTo T (closedBall 0 R ∩ P) (closedBall 0 R ∩ P) := by intro f hf exact ⟨hmaps hf.1, hP f hf.2 (mem_closedBall_zero_iff.mp hf.1)⟩ have hcontract : ContractingWith (1/2 : ℝ≥0) (hmapsP.restrict T _ _) := by refine ⟨by norm_num, LipschitzWith.of_dist_le_mul ?_⟩ intro f g change dist (T f.val) (T g.val) ≤ ((1/2 : ℝ≥0) : ℝ) * dist f.val g.val rw [dist_eq_norm, dist_eq_norm] exact hTd f.val g.val (mem_closedBall_zero_iff.mp f.property.1) (mem_closedBall_zero_iff.mp g.property.1) obtain ⟨f,hf,hfix,-⟩ := ContractingWith.exists_fixedPoint' (isClosed_closedBall.inter hPc).isComplete hmapsP hcontract ⟨mem_closedBall_zero_iff.mpr (by simpa only [norm_zero] using hR), hP0⟩ (edist_ne_top (0 : analyticDisk) (T 0)) refine ⟨f,hf.2,mem_closedBall_zero_iff.mp hf.1,?_,?_⟩ · have he : extend (f : DiskMap) = (r : ℂ) • extend (inverse c (H f) : DiskMap) := by calc extend (f : DiskMap) = extend (T f : DiskMap) := congrArg (fun g : analyticDisk => extend (g : DiskMap)) hfix.eq.symm _ = (r : ℂ) • extend (inverse c (H f) : DiskMap) := extend_smul _ _ rw [he, Pi.smul_apply, inverse_zero, smul_zero] · intro z hz have he : extend (f : DiskMap) = (r : ℂ) • extend (inverse c (H f) : DiskMap) := by calc extend (f : DiskMap) = extend (T f : DiskMap) := congrArg (fun g : analyticDisk => extend (g : DiskMap)) hfix.eq.symm _ = (r : ℂ) • extend (inverse c (H f) : DiskMap) := extend_smul _ _ rw [he, deriv_const_smul _ (differentiableAt_extend (inverse c (H f)) hz)] simp only [Pi.smul_apply, smul_eq_mul] have h := inverse_ode hc (H f) hz calc z * ((r : ℂ) * deriv (extend (inverse c (H f) : DiskMap)) z) + (c : ℂ) * ((r : ℂ) * extend (inverse c (H f) : DiskMap) z) = (r : ℂ) * (z * deriv (extend (inverse c (H f) : DiskMap)) z + (c : ℂ) * extend (inverse c (H f) : DiskMap) z) := by ring _ = (r : ℂ) * z * extend (H f : DiskMap) z := by rw [h, mul_assoc] theorem exists_fuchsian_disk {c r R M L : ℝ} (hc : 0 ≤ c) (hr : 0 ≤ r) (hR : 0 ≤ R) (hrM : r * M ≤ R) (hrL : r * L ≤ 1/2) (H : analyticDisk → analyticDisk) (hbound : ∀ f, ‖f‖ ≤ R → ‖H f‖ ≤ M) (hlip : ∀ f g, ‖f‖ ≤ R → ‖g‖ ≤ R → ‖H f - H g‖ ≤ L * ‖f-g‖) : ∃ f : analyticDisk, ‖f‖ ≤ R ∧ extend (f : DiskMap) 0 = 0 ∧ ∀ z ∈ ball (0 : ℂ) 1, z * deriv (extend (f : DiskMap)) z + (c : ℂ) * extend (f : DiskMap) z = (r : ℂ) * z * extend (H f : DiskMap) z := by obtain ⟨f,-,hf⟩ := exists_fuchsian_disk_invariant hc hr hR hrM hrL H hbound hlip univ isClosed_univ (mem_univ _) (fun _ _ _ => mem_univ _) exact ⟨f,hf⟩ theorem inputBound {r R : ℝ} (hr : 0 ≤ r) (hrR : r ≤ R) (f : analyticDisk) (hf : ‖f‖ ≤ R) (z : Disk) : ((r : ℂ) * (z : ℂ), (f : DiskMap) z) ∈ closedBall (0 : ℂ × ℂ) R := by rw [mem_closedBall_zero_iff, Prod.norm_def, max_le_iff] refine ⟨?_, ((f : DiskMap).norm_coe_le_norm z).trans hf⟩ rw [norm_mul, Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg hr] exact (mul_le_mul_of_nonneg_left (mem_closedBall_zero_iff.mp z.property) hr).trans (by simpa only [mul_one] using hrR) def composeMap {r R : ℝ} (hr : 0 ≤ r) (hrR : r ≤ R) (h : ℂ × ℂ → ℂ) (hcont : ContinuousOn h (closedBall 0 R)) (f : analyticDisk) (hf : ‖f‖ ≤ R) : DiskMap := ⟨fun z => h ((r : ℂ) * (z : ℂ), (f : DiskMap) z), hcont.comp_continuous ((continuous_const.mul continuous_subtype_val).prodMk (f : DiskMap).continuous) (inputBound hr hrR f hf)⟩ theorem extend_composeMap {r R : ℝ} (hr : 0 ≤ r) (hrR : r ≤ R) (h : ℂ × ℂ → ℂ) (hcont : ContinuousOn h (closedBall 0 R)) (f : analyticDisk) (hf : ‖f‖ ≤ R) {z : ℂ} (hz : z ∈ ball (0 : ℂ) 1) : extend (composeMap hr hrR h hcont f hf) z = h ((r : ℂ) * z, extend (f : DiskMap) z) := by have hz' := ball_subset_closedBall hz simp only [extend, dite_eq_left hz'] rfl def composeDisk {r R : ℝ} (hr : 0 ≤ r) (hrR : r ≤ R) (h : ℂ × ℂ → ℂ) (hdiff : DifferentiableOn ℂ h (closedBall 0 R)) (f : analyticDisk) (hf : ‖f‖ ≤ R) : analyticDisk := ⟨composeMap hr hrR h hdiff.continuousOn f hf, by change DifferentiableOn ℂ (extend (composeMap hr hrR h hdiff.continuousOn f hf)) (ball 0 1) have hi : DifferentiableOn ℂ (fun z : ℂ => ((r : ℂ)*z, extend (f : DiskMap) z)) (ball 0 1) := (differentiableOn_id.const_mul (r : ℂ)).prodMk f.property have hb : MapsTo (fun z : ℂ => ((r : ℂ)*z, extend (f : DiskMap) z)) (ball 0 1) (closedBall 0 R) := by intro z hz have hb := inputBound hr hrR f hf ⟨z,ball_subset_closedBall hz⟩ rw [← extend_coe] at hb exact hb exact (hdiff.comp hi hb).congr (fun z hz => extend_composeMap hr hrR h hdiff.continuousOn f hf hz)⟩ def nonlinearDisk {r R : ℝ} (hr : 0 ≤ r) (hrR : r ≤ R) (h : ℂ × ℂ → ℂ) (hdiff : DifferentiableOn ℂ h (closedBall 0 R)) (f : analyticDisk) : analyticDisk := by classical exact if hf : ‖f‖ ≤ R then composeDisk hr hrR h hdiff f hf else 0 theorem nonlinearDisk_apply {r R : ℝ} (hr : 0 ≤ r) (hrR : r ≤ R) (h : ℂ × ℂ → ℂ) (hdiff : DifferentiableOn ℂ h (closedBall 0 R)) (f : analyticDisk) (hf : ‖f‖ ≤ R) (z : Disk) : (nonlinearDisk hr hrR h hdiff f : DiskMap) z = h ((r : ℂ)*(z : ℂ), (f : DiskMap) z) := by classical simp only [nonlinearDisk, dite_eq_left hf] rfl theorem norm_nonlinearDisk_le {r R : ℝ} (hr : 0 ≤ r) (hrR : r ≤ R) (hR : 0 ≤ R) (h : ℂ × ℂ → ℂ) (hdiff : DifferentiableOn ℂ h (closedBall 0 R)) {K : ℝ≥0} (hlip : LipschitzOnWith K h (closedBall 0 R)) (f : analyticDisk) (hf : ‖f‖ ≤ R) : ‖nonlinearDisk hr hrR h hdiff f‖ ≤ ‖h 0‖ + (K : ℝ)*R := by change ‖(nonlinearDisk hr hrR h hdiff f : DiskMap)‖ ≤ _ apply (ContinuousMap.norm_le _ (by positivity)).2 intro z rw [nonlinearDisk_apply hr hrR h hdiff f hf] have hl := hlip.dist_le_mul ((r : ℂ)*(z : ℂ), (f : DiskMap) z) (inputBound hr hrR f hf z) 0 (mem_closedBall_self hR) rw [dist_zero_right] at hl have hnorm := mem_closedBall_zero_iff.mp (inputBound hr hrR f hf z) calc ‖h ((r : ℂ)*(z : ℂ), (f : DiskMap) z)‖ ≤ dist (h ((r : ℂ)*(z : ℂ), (f : DiskMap) z)) (h 0) + ‖h 0‖ := by simpa only [dist_eq_norm, sub_add_cancel] using norm_add_le (h ((r : ℂ)*(z : ℂ), (f : DiskMap) z) - h 0) (h 0) _ ≤ (K : ℝ) * R + ‖h 0‖ := by have hb := hl.trans (mul_le_mul_of_nonneg_left hnorm K.coe_nonneg) linarith only [hb] _ = ‖h 0‖ + (K : ℝ) * R := add_comm _ _ theorem nonlinearDisk_lip {r R : ℝ} (hr : 0 ≤ r) (hrR : r ≤ R) (h : ℂ × ℂ → ℂ) (hdiff : DifferentiableOn ℂ h (closedBall 0 R)) {K : ℝ≥0} (hlip : LipschitzOnWith K h (closedBall 0 R)) (f g : analyticDisk) (hf : ‖f‖ ≤ R) (hg : ‖g‖ ≤ R) : ‖nonlinearDisk hr hrR h hdiff f - nonlinearDisk hr hrR h hdiff g‖ ≤ (K : ℝ) * ‖f-g‖ := by change ‖(nonlinearDisk hr hrR h hdiff f : DiskMap) - (nonlinearDisk hr hrR h hdiff g : DiskMap)‖ ≤ _ apply (ContinuousMap.norm_le _ (mul_nonneg K.coe_nonneg (norm_nonneg _))).2 intro z simp only [ContinuousMap.sub_apply, nonlinearDisk_apply hr hrR h hdiff f hf, nonlinearDisk_apply hr hrR h hdiff g hg] have hl := hlip.dist_le_mul ((r : ℂ)*(z : ℂ), (f : DiskMap) z) (inputBound hr hrR f hf z) ((r : ℂ)*(z : ℂ), (g : DiskMap) z) (inputBound hr hrR g hg z) rw [Prod.dist_eq, dist_self, max_eq_right (dist_nonneg), dist_eq_norm, dist_eq_norm] at hl exact hl.trans (mul_le_mul_of_nonneg_left (((f-g : analyticDisk) : DiskMap).norm_coe_le_norm z) K.coe_nonneg) theorem exists_scaled_fuchsian {c : ℝ} (hc : 0 ≤ c) (h : ℂ × ℂ → ℂ) (ha : AnalyticAt ℂ h 0) : ∃ r : ℝ, 0 < r ∧ ∃ f : analyticDisk, extend (f : DiskMap) 0 = 0 ∧ ∀ z ∈ ball (0 : ℂ) 1, z * deriv (extend (f : DiskMap)) z + (c : ℂ) * extend (f : DiskMap) z = (r : ℂ) * z * h ((r : ℂ)*z, extend (f : DiskMap) z) := by obtain ⟨K,t,ht,hKt⟩ := (ha.contDiffAt (n := 1)).exists_lipschitzOnWith have hnhds : t ∩ {p | AnalyticAt ℂ h p} ∈ 𝓝 (0 : ℂ × ℂ) := inter_mem ht ha.eventually_analyticAt obtain ⟨ε,hε,hεsub⟩ := Metric.mem_nhds_iff.mp hnhds let R : ℝ := ε/2 have hR : 0 < R := by dsimp [R]; linarith have hsub : closedBall (0 : ℂ × ℂ) R ⊆ t ∩ {p | AnalyticAt ℂ h p} := by intro p hp apply hεsub exact mem_ball.mpr ((mem_closedBall.mp hp).trans_lt (by dsimp [R]; linarith)) have hdiff : DifferentiableOn ℂ h (closedBall 0 R) := fun p hp => (hsub hp).2.differentiableAt.differentiableWithinAt have hK : LipschitzOnWith K h (closedBall 0 R) := hKt.mono (fun _ hp => (hsub hp).1) let M : ℝ := ‖h 0‖ + (K : ℝ)*R have hM : 0 ≤ M := by dsimp [M]; positivity let r : ℝ := min (R/(M+1)) (min R (1/(2*((K : ℝ)+1)))) have hr : 0 < r := by dsimp [r]; positivity have hrR : r ≤ R := (min_le_right _ _).trans (min_le_left _ _) have hrM' : r * (M+1) ≤ R := (le_div_iff₀ (by linarith : 0 < M+1)).mp (min_le_left _ _) have hrM : r * M ≤ R := by nlinarith have hrK' : r * (2*((K : ℝ)+1)) ≤ 1 := (le_div_iff₀ (by positivity : 0 < 2*((K : ℝ)+1))).mp ((min_le_right _ _).trans (min_le_right _ _)) have hrK : r * (K : ℝ) ≤ 1/2 := by nlinarith let H := nonlinearDisk hr.le hrR h hdiff obtain ⟨f,hf,hf0,hode⟩ := exists_fuchsian_disk hc hr.le hR.le hrM hrK H (fun f hf => norm_nonlinearDisk_le hr.le hrR hR.le h hdiff hK f hf) (fun f g hf hg => nonlinearDisk_lip hr.le hrR h hdiff hK f g hf hg) refine ⟨r,hr,f,hf0,?_⟩ intro z hz have he : extend (H f : DiskMap) z = h ((r : ℂ)*z, extend (f : DiskMap) z) := by have he := nonlinearDisk_apply hr.le hrR h hdiff f hf ⟨z,ball_subset_closedBall hz⟩ rw [← extend_coe, ← extend_coe] at he exact he rw [← he] exact hode z hz def realDisk : Set analyticDisk := {f | ∀ z : Disk, (z : ℂ).im = 0 → ((f : DiskMap) z).im = 0} theorem realDisk_closed : IsClosed realDisk := by simp only [realDisk, ofPred_forall] exact isClosed_iInter (fun z => isClosed_iInter (fun _ => isClosed_eq (Complex.continuous_im.comp (eval z).continuous) continuous_const)) theorem realDisk_zero : (0 : analyticDisk) ∈ realDisk := by intro z hz rfl theorem realDisk_real_smul {f : analyticDisk} (hf : f ∈ realDisk) (r : ℝ) : (r : ℂ) • f ∈ realDisk := by intro z hz change ((r : ℂ) * (f : DiskMap) z).im = 0 simp only [Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, hf z hz, mul_zero, zero_mul, add_zero] theorem realDisk_radialIntegral {f : analyticDisk} (hf : f ∈ realDisk) {c : ℝ} (hc : 0 ≤ c) : radialIntegral c f ∈ realDisk := by intro z hz let L : analyticDisk →L[ℝ] ℝ := Complex.imCLM.comp ((eval z).restrictScalars ℝ) have he : ((radialIntegral c f : DiskMap) z).im = ∫ t : ℝ in 0..1, L (radialIntegrand c f t) := by exact (L.intervalIntegral_comp_comm ((continuous_radialIntegrand hc f).intervalIntegrable (μ := volume) 0 1)).symm rw [he] have hzero : ∀ t, L (radialIntegrand c f t) = 0 := by intro t change ((radialIntegrand c f t : DiskMap) z).im = 0 have ht : ((diskScale (clip t) z : Disk) : ℂ).im = 0 := by change (((clip t : ℝ) : ℂ) * (z : ℂ)).im = 0 simp [hz] change ((t ^ c : ℝ) • (f : DiskMap) (diskScale (clip t) z)).im = 0 rw [Complex.smul_im, hf _ ht, smul_zero] simp only [hzero, intervalIntegral.integral_zero] theorem realDisk_inverse {f : analyticDisk} (hf : f ∈ realDisk) {c : ℝ} (hc : 0 ≤ c) : inverse c f ∈ realDisk := by intro z hz change ((z : ℂ) * (radialIntegral c f : DiskMap) z).im = 0 simp only [Complex.mul_im, realDisk_radialIntegral hf hc z hz, hz, mul_zero, zero_mul, add_zero] theorem exists_scaled_real_fuchsian {c : ℝ} (hc : 0 ≤ c) (h : ℂ × ℂ → ℂ) (ha : AnalyticAt ℂ h 0) (hreal : ∀ p : ℂ × ℂ, p.1.im = 0 → p.2.im = 0 → (h p).im = 0) : ∃ r : ℝ, 0 < r ∧ ∃ f : analyticDisk, f ∈ realDisk ∧ extend (f : DiskMap) 0 = 0 ∧ ∀ z ∈ ball (0 : ℂ) 1, z * deriv (extend (f : DiskMap)) z + (c : ℂ) * extend (f : DiskMap) z = (r : ℂ) * z * h ((r : ℂ)*z, extend (f : DiskMap) z) := by obtain ⟨K,t,ht,hKt⟩ := (ha.contDiffAt (n := 1)).exists_lipschitzOnWith have hnhds : t ∩ {p | AnalyticAt ℂ h p} ∈ 𝓝 (0 : ℂ × ℂ) := inter_mem ht ha.eventually_analyticAt obtain ⟨ε,hε,hεsub⟩ := Metric.mem_nhds_iff.mp hnhds let R : ℝ := ε/2 have hR : 0 < R := by dsimp [R]; linarith have hsub : closedBall (0 : ℂ × ℂ) R ⊆ t ∩ {p | AnalyticAt ℂ h p} := by intro p hp apply hεsub exact mem_ball.mpr ((mem_closedBall.mp hp).trans_lt (by dsimp [R]; linarith)) have hdiff : DifferentiableOn ℂ h (closedBall 0 R) := fun p hp => (hsub hp).2.differentiableAt.differentiableWithinAt have hK : LipschitzOnWith K h (closedBall 0 R) := hKt.mono (fun _ hp => (hsub hp).1) let M : ℝ := ‖h 0‖ + (K : ℝ)*R have hM : 0 ≤ M := by dsimp [M]; positivity let r : ℝ := min (R/(M+1)) (min R (1/(2*((K : ℝ)+1)))) have hr : 0 < r := by dsimp [r]; positivity have hrR : r ≤ R := (min_le_right _ _).trans (min_le_left _ _) have hrM' : r * (M+1) ≤ R := (le_div_iff₀ (by linarith : 0 < M+1)).mp (min_le_left _ _) have hrM : r * M ≤ R := by nlinarith have hrK' : r * (2*((K : ℝ)+1)) ≤ 1 := (le_div_iff₀ (by positivity : 0 < 2*((K : ℝ)+1))).mp ((min_le_right _ _).trans (min_le_right _ _)) have hrK : r * (K : ℝ) ≤ 1/2 := by nlinarith let H := nonlinearDisk hr.le hrR h hdiff obtain ⟨f,hfreal,hf,hf0,hode⟩ := exists_fuchsian_disk_invariant hc hr.le hR.le hrM hrK H (fun f hf => norm_nonlinearDisk_le hr.le hrR hR.le h hdiff hK f hf) (fun f g hf hg => nonlinearDisk_lip hr.le hrR h hdiff hK f g hf hg) realDisk realDisk_closed realDisk_zero (by intro f hf hfR apply realDisk_real_smul _ r apply realDisk_inverse _ hc intro z hz change ((nonlinearDisk hr.le hrR h hdiff f : DiskMap) z).im = 0 rw [nonlinearDisk_apply hr.le hrR h hdiff f hfR] exact hreal _ (by simp [hz]) (hf z hz)) refine ⟨r,hr,f,hfreal,hf0,?_⟩ intro z hz have he : extend (H f : DiskMap) z = h ((r : ℂ)*z, extend (f : DiskMap) z) := by have he := nonlinearDisk_apply hr.le hrR h hdiff f hf ⟨z,ball_subset_closedBall hz⟩ rw [← extend_coe, ← extend_coe] at he exact he rw [← he] exact hode z hz /-- Real analytic local solvability of the post-jet regular-singular equation. The radius is constructed and positive; existence is not an input hypothesis. -/ theorem exists_real_fuchsian_germ {c : ℝ} (hc : 0 ≤ c) (h : ℂ × ℂ → ℂ) (ha : AnalyticAt ℂ h 0) (hreal : ∀ p : ℂ × ℂ, p.1.im = 0 → p.2.im = 0 → (h p).im = 0) : ∃ F : ℂ → ℂ, AnalyticAt ℂ F 0 ∧ F 0 = 0 ∧ (∀ x : ℝ, (F x).im = 0) ∧ ∀ᶠ z in 𝓝 (0 : ℂ), z * deriv F z + (c : ℂ)*F z = z*h (z,F z) := by obtain ⟨r,hr,f,hfr,hf0,hode⟩ := exists_scaled_real_fuchsian hc h ha hreal have hr0 : (r : ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr (ne_of_gt hr) let F : ℂ → ℂ := fun z => extend (f : DiskMap) (z/(r : ℂ)) have hmem {z : ℂ} (hz : z ∈ ball (0 : ℂ) r) : z/(r : ℂ) ∈ ball (0 : ℂ) 1 := by rw [mem_ball_zero_iff] at hz ⊢ rw [norm_div, Complex.norm_real, Real.norm_eq_abs, abs_of_pos hr] exact (div_lt_one hr).mpr hz refine ⟨F,?_,?_,?_,?_⟩ · have han : AnalyticAt ℂ (extend (f : DiskMap)) (0/(r : ℂ)) := by simpa only [zero_div] using f.property.analyticAt (ball_mem_nhds (0 : ℂ) (by norm_num : (0:ℝ)<1)) exact han.comp (f := fun z : ℂ => z/(r : ℂ)) (x := 0) (analyticAt_id.div_const (c := (r : ℂ))) · simpa only [F, zero_div] using hf0 · intro x dsimp [F] unfold extend split · apply hfr simp · rfl · filter_upwards [ball_mem_nhds (0 : ℂ) hr] with z hz have hm := hmem hz have he : deriv F z = deriv (extend (f : DiskMap)) (z/(r : ℂ)) / (r : ℂ) := by simpa only [F, Function.comp_def, id_eq, one_div, div_eq_mul_inv, one_mul] using (((differentiableAt_extend f hm).hasDerivAt).comp z ((hasDerivAt_id z).div_const (r : ℂ))).deriv have hd := hode (z/(r : ℂ)) hm have hzdiv : (r : ℂ)*(z/(r : ℂ)) = z := by field_simp rw [hzdiv] at hd change z * deriv F z + (c : ℂ)*extend (f : DiskMap) (z/(r : ℂ)) = _ rw [he] (convert hd using 1; ring) end SepticProfile.Fuchsian /- Sonic jet algebra over an arbitrary commutative ring. -/ noncomputable section SourceNoncomp namespace SepticProfile.Formal open PowerSeries variable {K : Type*} [CommRing K] abbrev Series (K : Type*) := PowerSeries K def z : Series K := 1 + X def prefactor (sigma : K) : Series K := z * (1 - C sigma*z^2) def den (sigma : K) (u : Series K) : Series K := prefactor sigma * (1-u^2) def r (c : K) (u : Series K) : Series K := 1-C c*u^2 def t (kappa : K) (u : Series K) : Series K := C kappa*z*(1-u^2)+3*(z-u) def residual (sigma kappa c : K) (u : Series K) : Series K := den sigma u * derivative u - r c u * t kappa u def lin (sigma kappa c : K) (u : Series K) : Series K := -2*prefactor sigma*u*derivative u + 2*C c*u*t kappa u + r c u*(2*C kappa*z*u+3) lemma coefficient_shift_zero (N M : ℕ) (h : N < M) (p : Series K) : coeff N (p*X^M) = 0 := by rw [coeff_mul_X_pow'] simp only [not_le.mpr h, ite_false] lemma coefficient_shift_same (N : ℕ) (p : Series K) : coeff N (p*X^N) = constantCoeff p := by simpa using coeff_mul_X_pow p N 0 lemma coefficient_monomial_second_order (n : ℕ) (a : K) (p : Series K) : coeff (n+2) ((C a*X^(n+2))^2*p) = 0 := by rw [mul_pow, ← pow_mul] have he : (C a ^ 2 * X ^ ((n + 2) * 2)) * p = (C a ^ 2 * p) * X ^ ((n + 2) * 2) := by ring rw [he] exact coefficient_shift_zero _ _ (by omega) _ lemma derivative_monomial (n : ℕ) (a : K) : derivative (C a*X^(n+1)) = C ((n+1)*a)*X^n := by simp only [Derivation.leibniz, derivative_C, derivative_pow, derivative_X] simp only [smul_eq_mul, mul_zero, add_zero, mul_one, Nat.add_sub_cancel] rw [← map_natCast C, ← mul_assoc, ← map_mul] congr 2 push_cast ring lemma coefficient_monomial_derivative (n : ℕ) (a : K) (p : Series K) : coeff (n+2) ((C a*X^(n+2))*derivative (C a*X^(n+2))*p) = 0 := by rw [show n+2 = (n+1)+1 by omega, derivative_monomial] have he : C a * X ^ (n + 1 + 1) * (C ((↑(n + 1) + 1) * a) * X ^ (n + 1)) * p = (C a * C ((↑(n + 1) + 1)*a)*p)*X^((n+1+1)+(n+1)) := by rw [pow_add] ring rw [he] exact coefficient_shift_zero _ _ (by omega) _ lemma residual_linearization (sigma kappa c : K) (u d : Series K) : residual sigma kappa c (u+d) - residual sigma kappa c u = den sigma u * derivative d + lin sigma kappa c u * d + d^2 * (-prefactor sigma * derivative u + C c*t kappa u + 2*C c*u*(-2*C kappa*z*u-3) + r c u*C kappa*z + (-2*C c*u*C kappa*z + C c*(-2*C kappa*z*u-3))*d - C c*C kappa*z*d^2) + d*derivative d*(-prefactor sigma*(2*u+d)) := by simp only [residual, den, lin, r, t, map_add] ring lemma residual_monomial_perturbation (sigma kappa c : K) (u : Series K) (n : ℕ) (a : K) : coeff (n+2) (residual sigma kappa c (u+C a*X^(n+2))) - coeff (n+2) (residual sigma kappa c u) = coeff (n+2) (den sigma u * derivative (C a*X^(n+2))) + coeff (n+2) (lin sigma kappa c u * (C a*X^(n+2))) := by rw [← map_sub, residual_linearization] simp only [map_add, coefficient_monomial_second_order, coefficient_monomial_derivative, add_zero] lemma coeff_den_one (sigma : K) {u : Series K} {s : K} (hu0 : constantCoeff u = 1) (hu1 : coeff 1 u = s) : coeff 1 (den sigma u) = -2*(1-sigma)*s := by simp only [den, prefactor, z, coeff_one_mul, coeff_one_pow, map_sub, map_add, map_mul, map_pow, constantCoeff_C, hu0, hu1] norm_num ring lemma coeff_lin_zero (sigma kappa c : K) {u : Series K} {s : K} (hu0 : constantCoeff u = 1) (hu1 : coeff 1 u = s) : constantCoeff (lin sigma kappa c u) = -2*(1-sigma)*s + (1-c)*(2*kappa+3) := by have hd : constantCoeff (derivative u) = s := by rw [← coeff_zero_eq_constantCoeff_apply, coeff_derivative] simpa using hu1 simp only [lin, prefactor, z, r, t, map_add, map_sub, map_mul, map_neg, map_pow, constantCoeff_C, constantCoeff_one, constantCoeff_X, map_ofNat, hu0, hd] ring lemma coeff_den_derivative_monomial (sigma : K) (u : Series K) (s : K) (hu0 : constantCoeff u = 1) (hu1 : coeff 1 u = s) (n : ℕ) (a : K) : coeff (n+2) (den sigma u * derivative (C a*X^(n+2))) = ((n+2)*a)*(-2*(1-sigma)*s) := by rw [show n+2 = (n+1)+1 by omega, derivative_monomial] rw [show den sigma u * (C ((↑(n+1)+1)*a)*X^(n+1)) = C ((↑(n+1)+1)*a)*(den sigma u*X^(n+1)) by ring] rw [coeff_C_mul, show n+1+1 = 1+(n+1) by omega, coeff_mul_X_pow, coeff_den_one sigma hu0 hu1] push_cast ring lemma coeff_lin_monomial (sigma kappa c : K) (u : Series K) (s : K) (hu0 : constantCoeff u = 1) (hu1 : coeff 1 u = s) (n : ℕ) (a : K) : coeff (n+2) (lin sigma kappa c u * (C a*X^(n+2))) = a*(-2*(1-sigma)*s+(1-c)*(2*kappa+3)) := by rw [show lin sigma kappa c u * (C a*X^(n+2)) = C a*(lin sigma kappa c u*X^(n+2)) by ring] rw [coeff_C_mul, coefficient_shift_same, coeff_lin_zero sigma kappa c hu0 hu1] /-- The exact nth-jet coefficient includes all nonlinear terms in the normalization. Its multiplier vanishes at an integral slope ratio. -/ lemma coefficient_step (sigma kappa c : K) (u : Series K) (s : K) (hu0 : constantCoeff u = 1) (hu1 : coeff 1 u = s) (n : ℕ) (a : K) : coeff (n+2) (residual sigma kappa c (u+C a*X^(n+2))) = coeff (n+2) (residual sigma kappa c u) + ((1-c)*(2*kappa+3)-2*(1-sigma)*s*(n+3))*a := by have h := residual_monomial_perturbation sigma kappa c u n a rw [coeff_den_derivative_monomial sigma u s hu0 hu1, coeff_lin_monomial sigma kappa c u s hu0 hu1] at h linear_combination h /- Coefficient locality at the singular point. Although differentiating costs one coefficient, its multiplier has zero constant term, so the nth residual coefficient only depends on coefficients through n. -/ def Agree (n : ℕ) (u v : Series K) : Prop := ∀ i ≤ n, coeff i u = coeff i v lemma Agree.refl (n : ℕ) (u : Series K) : Agree n u u := fun _ _ => rfl lemma Agree.add {n : ℕ} {u v p q : Series K} (h : Agree n u v) (h' : Agree n p q) : Agree n (u+p) (v+q) := by intro i hi simp only [map_add, h i hi, h' i hi] lemma Agree.sub {n : ℕ} {u v p q : Series K} (h : Agree n u v) (h' : Agree n p q) : Agree n (u-p) (v-q) := by intro i hi simp only [map_sub, h i hi, h' i hi] lemma Agree.mul {n : ℕ} {u v p q : Series K} (h : Agree n u v) (h' : Agree n p q) : Agree n (u*p) (v*q) := by intro i hi simp only [coeff_mul] apply Finset.sum_congr rfl intro ab hab have hh := Finset.HasAntidiagonal.mem_antidiagonal.mp hab rw [h ab.1 (by omega), h' ab.2 (by omega)] lemma Agree.pow {n : ℕ} {u v : Series K} (h : Agree n u v) (k : ℕ) : Agree n (u^k) (v^k) := by induction k with | zero => simpa only [pow_zero] using Agree.refl n (1 : Series K) | succ k hk => simpa only [pow_succ] using hk.mul h lemma Agree.mul_derivative {n : ℕ} {u v p q : Series K} (h : Agree n u v) (h' : Agree n p q) (hp : constantCoeff p = 0) (hq : constantCoeff q = 0) : Agree n (p*derivative u) (q*derivative v) := by intro i hi simp only [coeff_mul] apply Finset.sum_congr rfl intro ab hab have hh := Finset.HasAntidiagonal.mem_antidiagonal.mp hab by_cases hz : ab.1 = 0 · simp only [hz, coeff_zero_eq_constantCoeff_apply, hp, hq, zero_mul] · rw [h' ab.1 (by omega), coeff_derivative, coeff_derivative, h (ab.2+1) (by omega)] lemma residual_agree (sigma kappa c : K) {n : ℕ} {u v : Series K} (h : Agree n u v) (hu : constantCoeff u = 1) (hv : constantCoeff v = 1) : Agree n (residual sigma kappa c u) (residual sigma kappa c v) := by have hn : Agree n (1-u^2) (1-v^2) := (Agree.refl n 1).sub (h.pow 2) have hd : Agree n (den sigma u) (den sigma v) := (Agree.refl n (prefactor sigma)).mul hn have hd0 : constantCoeff (den sigma u) = 0 := by simp [den, hu] have hd0' : constantCoeff (den sigma v) = 0 := by simp [den, hv] have hr : Agree n (r c u) (r c v) := (Agree.refl n 1).sub ((Agree.refl n (C c)).mul (h.pow 2)) have ht : Agree n (t kappa u) (t kappa v) := ((Agree.refl n (C kappa*z)).mul hn).add ((Agree.refl n 3).mul ((Agree.refl n z).sub h)) exact (h.mul_derivative hd hd0 hd0').sub (hr.mul ht) def before (n : ℕ) (u : Series K) : Series K := mk fun i => if i < n then coeff i u else 0 lemma coeff_before (n : ℕ) (u : Series K) (i : ℕ) : coeff i (before n u) = if i < n then coeff i u else 0 := by simp [before] lemma before_zero (n : ℕ) (u : Series K) (hn : 0 < n) : constantCoeff (before n u) = constantCoeff u := by rw [← coeff_zero_eq_constantCoeff_apply, coeff_before, ite_eq_left hn, coeff_zero_eq_constantCoeff_apply] lemma before_one (n : ℕ) (u : Series K) (hn : 1 < n) : coeff 1 (before n u) = coeff 1 u := by rw [coeff_before, ite_eq_left hn] lemma agree_before_next (n : ℕ) (u : Series K) : Agree n u (before n u + C (coeff n u)*X^n) := by intro i hi rw [map_add, coeff_before, coeff_C_mul, coeff_X_pow] by_cases h : i = n · subst i; simp · simp [h, show i < n by omega] lemma step_at (sigma kappa c : K) (u : Series K) (s : K) (hu0 : constantCoeff u = 1) (hu1 : coeff 1 u = s) (n : ℕ) : coeff (n+2) (residual sigma kappa c u) = coeff (n+2) (residual sigma kappa c (before (n+2) u)) + ((1-c)*(2*kappa+3)-2*(1-sigma)*s*(n+3))*coeff (n+2) u := by have h0 : constantCoeff (before (n+2) u) = 1 := by rw [before_zero _ _ (by omega), hu0] have h1 : coeff 1 (before (n+2) u) = s := by rw [before_one _ _ (by omega), hu1] have hv : constantCoeff (before (n+2) u+C (coeff (n+2) u)*X^(n+2)) = 1 := by simp [h0] rw [(residual_agree sigma kappa c (agree_before_next (n+2) u) hu0 hv) _ le_rfl, coefficient_step sigma kappa c (before (n+2) u) s h0 h1] end SepticProfile.Formal namespace SepticProfile.SourceJets open Polynomial variable {K : Type*} [Field K] /-- The literal normalized source polynomial residual, expanded about z=1. -/ def residualP (sigma kappa c : K) (u : Polynomial K) : Polynomial K := (1+X)*(1-C sigma*(1+X)^2)*(1-u^2)*u.derivative - (1-C c*u^2)*(C kappa*(1+X)*(1-u^2)+3*((1+X)-u)) theorem coe_residualP (sigma kappa c : K) (u : Polynomial K) : (residualP sigma kappa c u : PowerSeries K) = Formal.residual sigma kappa c u := by simp only [residualP, Formal.residual, Formal.den, Formal.prefactor, Formal.z, Formal.r, Formal.t, coe_sub, coe_mul, coe_add, coe_pow, coe_one, coe_X, coe_C, ← PowerSeries.derivative_coe] have h3 : ((3 : Polynomial K) : PowerSeries K) = 3 := by rw [show (3 : Polynomial K) = 1+1+1 by ring, coe_add, coe_add, coe_one] ring rw [h3] theorem coeff_residual_one (sigma kappa c : K) {u : PowerSeries K} {s : K} (hu0 : PowerSeries.constantCoeff u=1) (hu1 : PowerSeries.coeff 1 u=s) : PowerSeries.coeff 1 (Formal.residual sigma kappa c u) = -2*(1-sigma)*s^2+(1-c)*(2*kappa+3)*s-3*(1-c) := by have hd : PowerSeries.constantCoeff (PowerSeries.derivative u)=s := by rw [← PowerSeries.coeff_zero_eq_constantCoeff_apply, PowerSeries.coeff_derivative] simpa using hu1 simp only [Formal.residual, Formal.den, Formal.prefactor, Formal.z, Formal.r, Formal.t, map_sub, map_add, map_mul, map_pow, PowerSeries.coeff_one_mul, PowerSeries.coeff_one_pow, PowerSeries.constantCoeff_C, map_ofNat, hu0, hu1, hd] norm_num ring theorem constantCoeff_residual (sigma kappa c : K) {u : PowerSeries K} (hu0 : PowerSeries.constantCoeff u=1) : PowerSeries.constantCoeff (Formal.residual sigma kappa c u) = 0 := by simp [Formal.residual, Formal.den, Formal.prefactor, Formal.z, Formal.r, Formal.t, hu0] /-- Construct all required finite sonic jets at a nonresonant weak slope. There is no assumed formal series. The zero multiplier at a resonance is explicitly excluded; this is why the resonant parameter cannot be used as a selected analytic profile parameter. -/ theorem exists_finite_jet (sigma kappa c s : K) (hs : 2*(1-sigma)*s^2-(1-c)*(2*kappa+3)*s+3*(1-c)=0) (N : ℕ) (hne : ∀ n < N, (1-c)*(2*kappa+3)-2*(1-sigma)*s*(n+3) ≠ 0) : ∃ p : Polynomial K, p.coeff 0=1 ∧ p.coeff 1=s ∧ ∀ i ≤ N+1, (residualP sigma kappa c p).coeff i=0 := by induction N with | zero => let p : Polynomial K := 1+C s*X have hp0 : p.coeff 0=1 := by simp [p] have hp1 : p.coeff 1=s := by simp [p, Polynomial.coeff_one] have hp0' : PowerSeries.constantCoeff (p : PowerSeries K)=1 := by simpa using hp0 have hp1' : PowerSeries.coeff 1 (p : PowerSeries K)=s := by simpa using hp1 refine ⟨p,hp0,hp1,?_⟩ intro i hi have hi' : i=0 ∨ i=1 := by omega rcases hi' with rfl | rfl · rw [← Polynomial.coeff_coe, coe_residualP, PowerSeries.coeff_zero_eq_constantCoeff_apply] exact constantCoeff_residual sigma kappa c hp0' · rw [← Polynomial.coeff_coe, coe_residualP, coeff_residual_one sigma kappa c hp0' hp1'] linear_combination -hs | succ n ih => obtain ⟨p,hp0,hp1,hp⟩ := ih (fun i hi => hne i (by omega)) let d : K := (1-c)*(2*kappa+3)-2*(1-sigma)*s*(n+3) have hd : d ≠ 0 := hne n (by omega) let a : K := -(residualP sigma kappa c p).coeff (n+2)/d let q : Polynomial K := p+C a*X^(n+2) have hq0 : q.coeff 0=1 := by simp [q,hp0] have hq1 : q.coeff 1=s := by simp [q,hp1] have hp0' : PowerSeries.constantCoeff (p : PowerSeries K)=1 := by simpa using hp0 have hp1' : PowerSeries.coeff 1 (p : PowerSeries K)=s := by simpa using hp1 have hq0' : PowerSeries.constantCoeff (q : PowerSeries K)=1 := by simpa using hq0 have hag : Formal.Agree (n+1) (q : PowerSeries K) (p : PowerSeries K) := by intro i hi simp only [Polynomial.coeff_coe] change q.coeff i = p.coeff i simp [q, show ¬i=n+2 by omega] refine ⟨q,hq0,hq1,?_⟩ intro i hi by_cases hi' : i=n+2 · subst i have he := Formal.coefficient_step sigma kappa c (p : PowerSeries K) s hp0' hp1' n a have hq : (q : PowerSeries K) = (p : PowerSeries K)+PowerSeries.C a*PowerSeries.X^(n+2) := by simp [q] rw [← hq, ← coe_residualP, ← coe_residualP, Polynomial.coeff_coe, Polynomial.coeff_coe] at he rw [he] change (residualP sigma kappa c p).coeff (n+2) + d * a = 0 dsimp only [a] (field_simp [hd]; ring) · have he := Formal.residual_agree sigma kappa c hag hq0' hp0' i (by omega) rw [← coe_residualP, ← coe_residualP, Polynomial.coeff_coe, Polynomial.coeff_coe] at he exact he.trans (hp i (by omega)) theorem exists_finite_jet_divisible (sigma kappa c s : K) (hs : 2*(1-sigma)*s^2-(1-c)*(2*kappa+3)*s+3*(1-c)=0) (N : ℕ) (hne : ∀ n < N, (1-c)*(2*kappa+3)-2*(1-sigma)*s*(n+3) ≠ 0) : ∃ p : Polynomial K, p.coeff 0=1 ∧ p.coeff 1=s ∧ (X : Polynomial K)^(N+2) ∣ residualP sigma kappa c p := by obtain ⟨p,hp0,hp1,hp⟩ := exists_finite_jet sigma kappa c s hs N hne exact ⟨p,hp0,hp1,X_pow_dvd_iff.mpr (fun i hi => hp i (by omega))⟩ end SepticProfile.SourceJets /- Algebraic finite-jet subtraction for the sonic germ. -/ noncomputable section SourceNoncomp namespace SepticProfile.Remainder variable {K : Type*} [CommRing K] def pref (sigma x : K) : K := (1+x)*(1-sigma*(1+x)^2) def den (sigma x u : K) : K := pref sigma x*(1-u^2) def amp (c u : K) : K := 1-c*u^2 def trans (kappa x u : K) : K := kappa*(1+x)*(1-u^2)+3*((1+x)-u) def res (sigma kappa c x u du : K) : K := den sigma x u*du - amp c u*trans kappa x u def lin (sigma kappa c x u du : K) : K := -2*pref sigma x*u*du + 2*c*u*trans kappa x u + amp c u*(2*kappa*(1+x)*u+3) def quad (sigma kappa c x u du d : K) : K := -pref sigma x*du + c*trans kappa x u + 2*c*u*(-2*kappa*(1+x)*u-3) + amp c u*kappa*(1+x) + (-2*c*u*kappa*(1+x)+c*(-2*kappa*(1+x)*u-3))*d - c*kappa*(1+x)*d^2 def denDiff (sigma x u d : K) : K := -pref sigma x*(2*u+d) theorem den_expansion (sigma x u d : K) : den sigma x (u+d) = den sigma x u+d*denDiff sigma x u d := by unfold den denDiff ring theorem res_expansion (sigma kappa c x u du d dd : K) : res sigma kappa c x (u+d) (du+dd) = res sigma kappa c x u du + den sigma x (u+d)*dd + lin sigma kappa c x u du*d + d^2*quad sigma kappa c x u du d := by simp only [res, den, lin, quad, amp, trans] ring def denQuot (sigma x p1 : K) : K := pref sigma x*(-2*p1-x*p1^2) theorem den_factor (sigma x p1 : K) : den sigma x (1+x*p1) = x*denQuot sigma x p1 := by simp only [den, denQuot] ring def jetA (sigma x p p1 g : K) (m : ℕ) : K := denQuot sigma x p1 + x^(m+1)*g*denDiff sigma x p (x^(m+2)*g) def jetE (sigma kappa c rho x p dp s e g : K) (m : ℕ) : K := s+e*g+x^(m+1)*g^2*quad sigma kappa c x p dp (x^(m+2)*g)+ rho*x^m*g^2*denDiff sigma x p (x^(m+2)*g) /-- Exact cleared algebraic identity; no small-denominator approximation. -/ theorem jet_identity (sigma kappa c rho x p p1 dp s e g dg : K) (m : ℕ) (hp : p=1+x*p1) (hr : res sigma kappa c x p dp=x^(m+3)*s) (hl : rho*denQuot sigma x p1+lin sigma kappa c x p dp=x*e) : res sigma kappa c x (p+x^(m+2)*g) (dp+(m+2)*x^(m+1)*g+x^(m+2)*dg) = x^(m+2)*(jetA sigma x p p1 g m * (x*dg+((m+2)-rho)*g)+ x*jetE sigma kappa c rho x p dp s e g m) := by have hd : den sigma x p=x*denQuot sigma x p1 := by rw [hp, den_factor] rw [show dp+(m+2)*x^(m+1)*g+x^(m+2)*dg = dp+((m+2)*x^(m+1)*g+x^(m+2)*dg) by ring, res_expansion, hr, den_expansion, hd] have hl' : lin sigma kappa c x p dp=x*e-rho*denQuot sigma x p1 := by linear_combination hl rw [hl'] simp only [jetA, jetE, pow_succ] ring theorem jetA_zero (sigma p p1 g : K) (m : ℕ) : jetA sigma 0 p p1 g m = -2*(1-sigma)*p1 := by simp [jetA, denQuot, pref] ring end SepticProfile.Remainder namespace SepticProfile.SourceSonic open Polynomial def prefP (sigma : ℝ) : Polynomial ℝ := (1+X)*(1-C sigma*(1+X)^2) def denQuotP (sigma : ℝ) (p1 : Polynomial ℝ) : Polynomial ℝ := prefP sigma*(-2*p1-X*p1^2) def transP (kappa : ℝ) (p : Polynomial ℝ) : Polynomial ℝ := C kappa*(1+X)*(1-p^2)+3*((1+X)-p) def linP (sigma kappa c : ℝ) (p : Polynomial ℝ) : Polynomial ℝ := -2*prefP sigma*p*p.derivative+2*C c*p*transP kappa p+ (1-C c*p^2)*(2*C kappa*(1+X)*p+3) theorem split_polynomial {p : Polynomial ℝ} {s : ℝ} (hp0 : p.coeff 0=1) (hp1 : p.coeff 1=s) : ∃ p1 : Polynomial ℝ, p=1+X*p1 ∧ p1.coeff 0=s := by have hd : (X : Polynomial ℝ) ∣ p-1 := X_dvd_iff.mpr (by simp [hp0]) obtain ⟨p1,hp1'⟩ := hd have he : p=1+X*p1 := by linear_combination hp1' refine ⟨p1,he,?_⟩ rw [he] at hp1 simpa [coeff_one] using hp1 theorem coeff_denQuotP {p1 : Polynomial ℝ} {s : ℝ} (h : p1.coeff 0=s) (sigma : ℝ) : (denQuotP sigma p1).coeff 0 = -2*(1-sigma)*s := by rw [coeff_zero_eq_eval_zero] have he : p1.eval 0=s := by simpa only [coeff_zero_eq_eval_zero] using h simp [denQuotP, prefP, he] ring theorem coeff_linP {p : Polynomial ℝ} {s : ℝ} (hp0 : p.coeff 0=1) (hp1 : p.coeff 1=s) (sigma kappa c : ℝ) : (linP sigma kappa c p).coeff 0 = -2*(1-sigma)*s+(1-c)*(2*kappa+3) := by rw [coeff_zero_eq_eval_zero] have he0 : p.eval 0=1 := by simpa only [coeff_zero_eq_eval_zero] using hp0 have he1 : p.derivative.eval 0=s := by rw [← coeff_zero_eq_eval_zero, coeff_derivative] simpa using hp1 simp [linP, prefP, transP, he0, he1] theorem split_linearization {p p1 : Polynomial ℝ} {s rho sigma kappa c : ℝ} (hp0 : p.coeff 0=1) (hp1 : p.coeff 1=s) (hp10 : p1.coeff 0=s) (hrho : rho*(2*(1-sigma)*s)=(1-c)*(2*kappa+3)-2*(1-sigma)*s) : ∃ e : Polynomial ℝ, C rho*denQuotP sigma p1+linP sigma kappa c p=X*e := by apply X_dvd_iff.mpr rw [coeff_add, coeff_C_mul, coeff_denQuotP hp10, coeff_linP hp0 hp1] linear_combination -hrho theorem aeval_prefP (sigma : ℝ) (z : ℂ) : aeval z (prefP sigma)=Remainder.pref (sigma : ℂ) z := by simp [prefP, Remainder.pref] theorem aeval_denQuotP (sigma : ℝ) (p1 : Polynomial ℝ) (z : ℂ) : aeval z (denQuotP sigma p1)=Remainder.denQuot (sigma : ℂ) z (aeval z p1) := by simp [denQuotP, Remainder.denQuot, aeval_prefP, map_ofNat] theorem aeval_transP (kappa : ℝ) (p : Polynomial ℝ) (z : ℂ) : aeval z (transP kappa p)=Remainder.trans (kappa : ℂ) z (aeval z p) := by simp [transP, Remainder.trans, map_ofNat] theorem aeval_linP (sigma kappa c : ℝ) (p : Polynomial ℝ) (z : ℂ) : aeval z (linP sigma kappa c p)=Remainder.lin (sigma : ℂ) (kappa : ℂ) (c : ℂ) z (aeval z p) (aeval z p.derivative) := by simp [linP, Remainder.lin, Remainder.amp, aeval_prefP, aeval_transP, map_ofNat] theorem aeval_residualP (sigma kappa c : ℝ) (p : Polynomial ℝ) (z : ℂ) : aeval z (SourceJets.residualP sigma kappa c p)=Remainder.res (sigma : ℂ) (kappa : ℂ) (c : ℂ) z (aeval z p) (aeval z p.derivative) := by simp [SourceJets.residualP, Remainder.res, Remainder.den, Remainder.pref, Remainder.amp, Remainder.trans, map_ofNat] theorem aeval_real (p : Polynomial ℝ) {z : ℂ} (hz : z.im=0) : (aeval z p : ℂ).im=0 := by have he : z=(z.re : ℂ) := by apply Complex.ext <;> simp [hz] rw [he] have he' : aeval ((z.re : ℝ) : ℂ) p = ((p.eval z.re : ℝ) : ℂ) := aeval_algebraMap_apply_eq_algebraMap_eval (A := ℂ) _ _ rw [he'] rfl end SepticProfile.SourceSonic namespace SepticProfile.SourceSonic open Polynomial Set Filter open scoped Topology def sourceH (sigma kappa c rho : ℝ) (p p1 S E : Polynomial ℝ) (m : ℕ) (v : ℂ × ℂ) : ℂ := -Remainder.jetE (sigma : ℂ) (kappa : ℂ) (c : ℂ) (rho : ℂ) v.1 (aeval v.1 p) (aeval v.1 p.derivative) (aeval v.1 S) (aeval v.1 E) v.2 m / Remainder.jetA (sigma : ℂ) v.1 (aeval v.1 p) (aeval v.1 p1) v.2 m theorem sourceH_analytic (sigma kappa c rho s : ℝ) (p p1 S E : Polynomial ℝ) (m : ℕ) (hp10 : p1.coeff 0=s) (hd : 2*(1-sigma)*s≠0) : AnalyticAt ℂ (sourceH sigma kappa c rho p p1 S E m) 0 := by have hf (a : ℂ × ℂ) : AnalyticAt ℂ (fun v : ℂ × ℂ => v.1) a := analyticAt_fst have hg (a : ℂ × ℂ) : AnalyticAt ℂ (fun v : ℂ × ℂ => v.2) a := analyticAt_snd have han (q : Polynomial ℝ) (a : ℂ × ℂ) : AnalyticAt ℂ (fun v : ℂ × ℂ => aeval v.1 q) a := (hf a).aeval_polynomial q have hden : Remainder.jetA (sigma : ℂ) (0 : ℂ × ℂ).1 (aeval (0 : ℂ × ℂ).1 p) (aeval (0 : ℂ × ℂ).1 p1) (0 : ℂ × ℂ).2 m ≠ 0 := by rw [Prod.fst_zero, Prod.snd_zero, Remainder.jetA_zero, ← coeff_zero_eq_aeval_zero', hp10] change -2*(1-(sigma : ℂ))*(s : ℂ) ≠ 0 have hdc : (2 : ℂ)*(1-(sigma : ℂ))*(s : ℂ) ≠ 0 := by exact_mod_cast hd simpa only [neg_mul] using neg_ne_zero.mpr hdc unfold sourceH apply AnalyticAt.div · simp only [Remainder.jetE, Remainder.quad, Remainder.pref, Remainder.trans, Remainder.amp, Remainder.denDiff] fun_prop · simp only [Remainder.jetA, Remainder.denQuot, Remainder.pref, Remainder.denDiff] fun_prop · exact hden theorem sourceH_real (sigma kappa c rho : ℝ) (p p1 S E : Polynomial ℝ) (m : ℕ) (v : ℂ × ℂ) (hx : v.1.im=0) (hy : v.2.im=0) : (sourceH sigma kappa c rho p p1 S E m v).im=0 := by apply Complex.conj_eq_iff_im.mp have ha (q : Polynomial ℝ) : starRingEnd ℂ (aeval v.1 q)=aeval v.1 q := Complex.conj_eq_iff_im.mpr (aeval_real q hx) have hx' := Complex.conj_eq_iff_im.mpr hx have hy' := Complex.conj_eq_iff_im.mpr hy simp only [sourceH, Remainder.jetE, Remainder.jetA, Remainder.quad, Remainder.pref, Remainder.trans, Remainder.amp, Remainder.denDiff, Remainder.denQuot, map_add, map_sub, map_mul, map_pow, map_neg, map_div₀, map_ofNat, map_one, Complex.conj_ofReal, hx', hy', ha] end SepticProfile.SourceSonic namespace SepticProfile.SourceSonic open Polynomial Set Filter open scoped Topology theorem derivative_jet_add (p : Polynomial ℝ) (F : ℂ → ℂ) (m : ℕ) {z : ℂ} (hF : DifferentiableAt ℂ F z) : HasDerivAt (fun x : ℂ => aeval x p+x^(m+2)*F x) (aeval z p.derivative+((m : ℂ)+2)*z^(m+1)*F z+z^(m+2)*deriv F z) z := by convert (p.hasDerivAt_aeval z).add (((hasDerivAt_id z).pow (m+2)).mul hF.hasDerivAt) using 1 all_goals first | rfl | (simp [add_assoc]) /-- A nonresonant analytic transonic germ for the normalized residual. -/ theorem exists_sonic_germ (sigma kappa c s rho : ℝ) (hs : 2*(1-sigma)*s^2-(1-c)*(2*kappa+3)*s+3*(1-c)=0) (hd : 2*(1-sigma)*s ≠ 0) (hrho : rho*(2*(1-sigma)*s)=(1-c)*(2*kappa+3)-2*(1-sigma)*s) (hnr : ∀ n : ℕ, rho ≠ (n : ℝ)+2) : ∃ U : ℂ → ℂ, AnalyticAt ℂ U 0 ∧ U 0=1 ∧ deriv U 0=(s : ℂ) ∧ (∀ x : ℝ, (U x).im=0) ∧ ∀ᶠ z in 𝓝 (0 : ℂ), (1+z)*(1-(sigma : ℂ)*(1+z)^2)*(1-(U z)^2)*deriv U z = (1-(c : ℂ)*(U z)^2)*((kappa : ℂ)*(1+z)*(1-(U z)^2)+3*((1+z)-U z)) := by obtain ⟨m,hm⟩ := exists_nat_gt rho have hcoef (n : ℕ) : (1-c)*(2*kappa+3)-2*(1-sigma)*s*((n : ℝ)+3) ≠ 0 := by intro he have hz : (2*(1-sigma)*s)*(rho-((n : ℝ)+2))=0 := by linear_combination hrho + he exact hnr n (sub_eq_zero.mp ((mul_eq_zero.mp hz).resolve_left hd)) obtain ⟨p,hp0,hp1,S,hS⟩ := SourceJets.exists_finite_jet_divisible sigma kappa c s hs (m+1) (fun n _ => hcoef n) have hS' : SourceJets.residualP sigma kappa c p=X^(m+3)*S := by simpa only [Nat.add_assoc] using hS obtain ⟨p1,hp,hp10⟩ := split_polynomial hp0 hp1 obtain ⟨E,hE⟩ := split_linearization hp0 hp1 hp10 hrho have hc : 0 ≤ ((m : ℝ)+2)-rho := by linarith obtain ⟨F,hFa,hF0,hFr,hFe⟩ := Fuchsian.exists_real_fuchsian_germ hc (sourceH sigma kappa c rho p p1 S E m) (sourceH_analytic sigma kappa c rho s p p1 S E m hp10 hd) (sourceH_real sigma kappa c rho p p1 S E m) let U : ℂ → ℂ := fun z => aeval z p+z^(m+2)*F z have han : AnalyticAt ℂ U 0 := (analyticAt_id.aeval_polynomial p).add ((analyticAt_id.pow (m+2)).mul hFa) have hdu {z : ℂ} (hz : DifferentiableAt ℂ F z) : deriv U z=aeval z p.derivative+((m : ℂ)+2)*z^(m+1)*F z+z^(m+2)*deriv F z := (derivative_jet_add p F m hz).deriv have hdc : (2 : ℂ)*(1-(sigma : ℂ))*(s : ℂ) ≠ 0 := by exact_mod_cast hd let A : ℂ → ℂ := fun z => Remainder.jetA (sigma : ℂ) z (aeval z p) (aeval z p1) (F z) m have hAc : ContinuousAt A 0 := by have hf := hFa.continuousAt have hpq (q : Polynomial ℝ) : ContinuousAt (fun z : ℂ => aeval z q) 0 := ((analyticAt_id : AnalyticAt ℂ (fun z : ℂ => z) 0).aeval_polynomial q).continuousAt dsimp only [A, Remainder.jetA, Remainder.denQuot, Remainder.denDiff, Remainder.pref] fun_prop have hA0 : A 0 ≠ 0 := by simp only [A, Remainder.jetA_zero, ← coeff_zero_eq_aeval_zero', hp10] change -2*(1-(sigma : ℂ))*(s : ℂ) ≠ 0 simpa only [neg_mul] using neg_ne_zero.mpr hdc refine ⟨U,han,?_,?_,?_,?_⟩ · simp only [U, zero_pow (by omega : m+2≠0), zero_mul, add_zero, ← coeff_zero_eq_aeval_zero', hp0, map_one] · rw [hdu hFa.differentiableAt] simp only [zero_pow (by omega : m+1≠0), zero_pow (by omega : m+2≠0), mul_zero, zero_mul, add_zero, ← coeff_zero_eq_aeval_zero', coeff_derivative] simp only [Nat.cast_zero, zero_add, mul_one, hp1] rfl · intro x apply Complex.conj_eq_iff_im.mp have hpconj := Complex.conj_eq_iff_im.mpr (aeval_real p (z := (x : ℂ)) (by simp)) have hfconj := Complex.conj_eq_iff_im.mpr (hFr x) simp only [U, map_add, map_mul, map_pow, hpconj, hfconj, Complex.conj_ofReal] · filter_upwards [hFe,hFa.eventually_analyticAt,hAc.eventually_ne hA0] with z hfe hfz hAz have hpz : aeval z p=1+z*aeval z p1 := by rw [hp]; simp have hsz : Remainder.res (sigma : ℂ) (kappa : ℂ) (c : ℂ) z (aeval z p) (aeval z p.derivative)=z^(m+3)*aeval z S := by rw [← aeval_residualP, hS']; simp have hez : (rho : ℂ)*Remainder.denQuot (sigma : ℂ) z (aeval z p1)+ Remainder.lin (sigma : ℂ) (kappa : ℂ) (c : ℂ) z (aeval z p) (aeval z p.derivative)= z*aeval z E := by rw [← aeval_denQuotP, ← aeval_linP] have he := congrArg (aeval z : Polynomial ℝ →ₐ[ℝ] ℂ) hE simpa using he have heq := Remainder.jet_identity (sigma : ℂ) (kappa : ℂ) (c : ℂ) (rho : ℂ) z (aeval z p) (aeval z p1) (aeval z p.derivative) (aeval z S) (aeval z E) (F z) (deriv F z) m hpz hsz hez have hcanc : A z*(z*deriv F z+((m : ℂ)+2-(rho : ℂ))*F z)+ z*Remainder.jetE (sigma : ℂ) (kappa : ℂ) (c : ℂ) (rho : ℂ) z (aeval z p) (aeval z p.derivative) (aeval z S) (aeval z E) (F z) m=0 := by push_cast at hfe rw [hfe] change A z*(z*(-Remainder.jetE _ _ _ _ _ _ _ _ _ _ _ / A z))+_ = 0 (field_simp [hAz]; ring) have hres : Remainder.res (sigma : ℂ) (kappa : ℂ) (c : ℂ) z (U z) (deriv U z)=0 := by rw [hdu hfz.differentiableAt] change Remainder.res _ _ _ _ (aeval z p+z^(m+2)*F z) _=0 rw [heq] change z^(m+2)*(A z*(z*deriv F z+((m : ℂ)+2-(rho : ℂ))*F z)+_)=0 rw [hcanc, mul_zero] exact sub_eq_zero.mp hres end SepticProfile.SourceSonic namespace SepticProfile.SourceSonic open Filter open scoped Topology ContDiff theorem exists_real_sonic_germ (sigma kappa c s rho : ℝ) (hs : 2*(1-sigma)*s^2-(1-c)*(2*kappa+3)*s+3*(1-c)=0) (hd : 2*(1-sigma)*s ≠ 0) (hrho : rho*(2*(1-sigma)*s)=(1-c)*(2*kappa+3)-2*(1-sigma)*s) (hnr : ∀ n : ℕ, rho ≠ (n : ℝ)+2) : ∃ u : ℝ → ℝ, AnalyticAt ℝ u 0 ∧ u 0=1 ∧ deriv u 0=s ∧ ∀ᶠ x in 𝓝 (0 : ℝ), (1+x)*(1-sigma*(1+x)^2)*(1-(u x)^2)*deriv u x = (1-c*(u x)^2)*(kappa*(1+x)*(1-(u x)^2)+3*((1+x)-u x)) := by obtain ⟨U,hUa,hU0,hUd,hUr,hUe⟩ := exists_sonic_germ sigma kappa c s rho hs hd hrho hnr let u : ℝ → ℝ := fun x => (U x).re have hucont : AnalyticAt ℝ u 0 := by exact (show AnalyticAt ℂ U (0 : ℝ) by simpa only [Complex.ofReal_zero] using hUa).re_ofReal have hdu {x : ℝ} (hx : DifferentiableAt ℂ U x) : deriv u x=(deriv U (x : ℂ)).re := hx.hasDerivAt.real_of_complex.deriv refine ⟨u,hucont,?_,?_,?_⟩ · simpa only [u, Complex.ofReal_zero, hU0] using (Complex.one_re : (1 : ℂ).re=1) · rw [hdu (by simpa using hUa.differentiableAt)] simp [hUd] · have ht : Tendsto (fun x : ℝ => (x : ℂ)) (𝓝 0) (𝓝 (0 : ℂ)) := by simpa only [ContinuousAt, Complex.ofReal_zero] using (Complex.continuous_ofReal.continuousAt (x := (0 : ℝ))) filter_upwards [ht.eventually hUe,ht.eventually hUa.eventually_analyticAt] with x hxe hxa have huv : U (x : ℂ)=(u x : ℂ) := by apply Complex.ext <;> simp [u,hUr] have he := congrArg Complex.re hxe rw [huv] at he simp only [← Complex.ofReal_one, ← Complex.ofReal_add, ← Complex.ofReal_mul, ← Complex.ofReal_sub, ← Complex.ofReal_pow, ← Complex.ofReal_ofNat] at he rw [hdu hxa.differentiableAt] simpa only [Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, zero_mul, sub_zero] using he end SepticProfile.SourceSonic namespace SepticProfile.SourceEuler open Filter open scoped Topology def changeOfVariable (h t : ℝ) : ℝ := -h*t/(1-t) def A (t : ℝ) : ℝ := 1-t def B (h t : ℝ) : ℝ := 1-(1+h)*t def residual (sigma kappa c h t u du : ℝ) : ℝ := B h t*((A t)^2-sigma*(B h t)^2)*(1-u^2)*du + h*(1-c*u^2)*(kappa*B h t*(1-u^2)+3*(B h t-A t*u)) lemma derivative_changeOfVariable (h t : ℝ) (ht : 1-t≠0) : HasDerivAt (changeOfVariable h) (-h/(1-t)^2) t := by have hd := (((hasDerivAt_id t).const_mul (-h)).div ((hasDerivAt_const t 1).sub (hasDerivAt_id t)) ht) convert hd using 1 all_goals first | rfl | (dsimp [changeOfVariable]; congr 1; ring) lemma transform_equation (sigma kappa c h t u du : ℝ) (ht : 1-t≠0) (he : (1+changeOfVariable h t)*(1-sigma*(1+changeOfVariable h t)^2)*(1-u^2)*du = (1-c*u^2)*(kappa*(1+changeOfVariable h t)*(1-u^2)+3*((1+changeOfVariable h t)-u))) : residual sigma kappa c h t u (du*(-h/(1-t)^2))=0 := by unfold changeOfVariable at he unfold residual A B field_simp at he ⊢ linear_combination -h*he theorem exists_real_euler_germ (sigma kappa c h s rho : ℝ) (hs : 2*(1-sigma)*s^2-(1-c)*(2*kappa+3)*s+3*(1-c)=0) (hd : 2*(1-sigma)*s ≠ 0) (hrho : rho*(2*(1-sigma)*s)=(1-c)*(2*kappa+3)-2*(1-sigma)*s) (hnr : ∀ n : ℕ, rho ≠ (n : ℝ)+2) : ∃ u : ℝ → ℝ, AnalyticAt ℝ u 0 ∧ u 0=1 ∧ deriv u 0=-h*s ∧ ∀ᶠ t in 𝓝 (0 : ℝ), residual sigma kappa c h t (u t) (deriv u t)=0 := by obtain ⟨v,hva,hv0,hvd,hve⟩ := SepticProfile.SourceSonic.exists_real_sonic_germ sigma kappa c s rho hs hd hrho hnr have hp0 : changeOfVariable h 0=0 := by simp [changeOfVariable] have hpa : AnalyticAt ℝ (changeOfVariable h) 0 := by unfold changeOfVariable fun_prop (disch := norm_num) let u := v ∘ changeOfVariable h have hua : AnalyticAt ℝ u 0 := by exact (hp0 ▸ hva).comp hpa have huder {t : ℝ} (ht : 1-t≠0) (hvt : DifferentiableAt ℝ v (changeOfVariable h t)) : deriv u t=deriv v (changeOfVariable h t)*(-h/(1-t)^2) := by exact ((hvt.hasDerivAt).comp t (derivative_changeOfVariable h t ht)).deriv refine ⟨u,hua,by simpa [u,hp0] using hv0,?_,?_⟩ · rw [huder (by norm_num) (by simpa only [hp0] using hva.differentiableAt),hp0,hvd] ring · have hpt : Tendsto (changeOfVariable h) (𝓝 0) (𝓝 0) := by simpa only [ContinuousAt,hp0] using hpa.continuousAt have htn : ∀ᶠ t : ℝ in 𝓝 0, 1-t≠0 := by exact (continuous_const.sub continuous_id).continuousAt.eventually_ne (by norm_num) filter_upwards [hpt.eventually hve,hpt.eventually hva.eventually_analyticAt,htn] with t hte hta htn rw [huder htn hta.differentiableAt] exact transform_equation sigma kappa c h t (v (changeOfVariable h t)) (deriv v (changeOfVariable h t)) htn hte end SepticProfile.SourceEuler end SourceNoncomp end SourceNoncomp end SourceNoncomp end SourceConstruction section ParameterBounds /-! A rational chart on the sonic quadratic. The interval avoids integral sonic resonances and satisfies the required parameter inequalities. -/ noncomputable section ParameterNoncomp namespace SepticProfile.ShootingParameters def leftEnd : ℝ := -706019/1000000 def rightEnd : ℝ := -705986/1000000 def a (t : ℝ) : ℝ := 27*t^2-5 def d (t : ℝ) : ℝ := 81*t^2-18*t-20 def e (t : ℝ) : ℝ := 81*t^2-9*t-21 def k (t : ℝ) : ℝ := 162*t^2+18*t-67 def beta (t : ℝ) : ℝ := 9*(1-9*t)/(2*d t) def q (t : ℝ) : ℝ := 3*e t/d t def sigma (t : ℝ) : ℝ := 3*(d t)^2/(5*(e t)^2) def kappa (t : ℝ) : ℝ := (162*t^2+45*t-49)/(2*e t) def slope (t : ℝ) : ℝ := e t/(3*a t) def ratio (t : ℝ) : ℝ := 9*a t/k t theorem rough_bounds {t : ℝ} (ht : t ∈ Set.Icc leftEnd rightEnd) : -(71/100 : ℝ) linarith [ht.1,ht.2] theorem a_pos {t : ℝ} (ht : t ∈ Set.Icc leftEnd rightEnd) : 0 nlinarith theorem nonresonance {t : ℝ} (ht : t ∈ Set.Icc leftEnd rightEnd) : ∀ n : ℕ, ratio t ≠ (n : ℝ)+2 := by intro n hn have h := ratio_bounds ht rw [hn] at h have hn1 : 71 < n := by exact_mod_cast (by linarith [h.1] : (71:ℝ)<(n:ℝ)) have hn2 : n < 72 := by exact_mod_cast (by linarith [h.2] : (n:ℝ)<(72:ℝ)) omega theorem beta_bounds {t : ℝ} (ht : t ∈ Set.Icc leftEnd rightEnd) : 1 < beta t ∧ beta t < 1001/1000 := by have hd := d_pos ht have hr := rough_bounds ht have hl : 0≤(t-leftEnd)*(-162*(t+leftEnd)-45) := by apply mul_nonneg (sub_nonneg.mpr ht.1) dsimp [leftEnd] linarith [hr.2] have hu : 0≤(rightEnd-t)*(-162162*(t+rightEnd)-44964) := by apply mul_nonneg (sub_nonneg.mpr ht.2) dsimp [rightEnd] linarith [hr.2] rw [beta,lt_div_iff₀ (by positivity),div_lt_iff₀ (by positivity)] dsimp [d,leftEnd,rightEnd] at * constructor <;> nlinarith theorem sigma_bounds {t : ℝ} (ht : t ∈ Set.Icc leftEnd rightEnd) : 0 < sigma t ∧ sigma t < 1 := by have hd := d_pos ht have he := e_pos ht have hb := beta_bounds ht have hi := parameter_identities ht have hq : 233/100 < q t := by rw [hi.1]; linarith [hb.2] constructor · dsimp [sigma] positivity · rw [hi.2.1,div_lt_one (by positivity)] nlinarith [sq_nonneg (q t-233/100)] theorem slope_pos {t : ℝ} (ht : t ∈ Set.Icc leftEnd rightEnd) : 0 < slope t := div_pos (e_pos ht) (mul_pos (by norm_num) (a_pos ht)) theorem sonic_divisor_ne {t : ℝ} (ht : t ∈ Set.Icc leftEnd rightEnd) : 2*(1-sigma t)*slope t ≠ 0 := ne_of_gt (mul_pos (mul_pos (by norm_num) (sub_pos.mpr (sigma_bounds ht).2)) (slope_pos ht)) theorem ratio_identity {t : ℝ} (ht : t ∈ Set.Icc leftEnd rightEnd) : ratio t*(2*(1-sigma t)*slope t) = (1-(3/5:ℝ))*(2*kappa t+3)-2*(1-sigma t)*slope t := by have ha := ne_of_gt (a_pos ht) have he := ne_of_gt (e_pos ht) have hk := ne_of_gt (k_pos ht) simp only [ratio,sigma,kappa,slope] field_simp dsimp [a,d,e,k] ring theorem standing_beta_bound {t : ℝ} (ht : t ∈ Set.Icc leftEnd rightEnd) : 1 < beta t ∧ beta t*((5/3:ℝ)+Real.sqrt (5/3:ℝ)) < 3 := by have hb := beta_bounds ht have hs : Real.sqrt (5/3:ℝ)<13/10 := by have he := Real.sq_sqrt (by norm_num : (0:ℝ)≤5/3) have hn := Real.sqrt_nonneg (5/3:ℝ) nlinarith refine ⟨hb.1,?_⟩ have hm := mul_lt_mul_of_pos_left hs (by linarith [hb.1] : 0 < beta t) nlinarith [hb.2] theorem shooting_coefficient_bounds {t : ℝ} (ht : t ∈ Set.Icc leftEnd rightEnd) : 991/1000 < sigma t ∧ sigma t < 993/1000 ∧ -(1/500:ℝ) < kappa t ∧ kappa t < 0 := by have hb := beta_bounds ht have hi := parameter_identities ht have hq1 : (3499/1500:ℝ) < q t := by rw [hi.1]; linarith [hb.2] have hq2 : q t < (7/3:ℝ) := by rw [hi.1]; linarith [hb.1] have hq : 0 < q t := by linarith have hqlo : (3499/1500:ℝ)^2 < (q t)^2 := by nlinarith [sq_nonneg (q t-3499/1500)] have hqhi : (q t)^2 < (7/3:ℝ)^2 := by nlinarith [mul_pos (sub_pos.mpr hq2) (by linarith : 0 < 7/3+q t)] rw [hi.2.1,hi.2.2] constructor · rw [lt_div_iff₀ (by positivity)] nlinarith constructor · rw [div_lt_iff₀ (by positivity)] nlinarith constructor · rw [lt_div_iff₀ hq] linarith [hb.2] · exact div_neg_of_neg_of_pos (by linarith [hb.1]) hq end SepticProfile.ShootingParameters end ParameterNoncomp end ParameterBounds /-! The sonic formal series is the Taylor jet of a smooth profile; this construction does not assume analyticity. -/ noncomputable section namespace SepticProfile.SonicJet open PowerSeries Filter open scoped Topology ContDiff abbrev Series := PowerSeries ℝ def jet (f : ℝ → ℝ) : Series := mk fun n => iteratedDeriv n f 0 / n.factorial lemma coeff_jet (f : ℝ → ℝ) (n : ℕ) : coeff n (jet f) = iteratedDeriv n f 0 / n.factorial := by simp [jet] lemma jet_congr {f g : ℝ → ℝ} (h : f =ᶠ[𝓝 0] g) : jet f = jet g := by ext n rw [coeff_jet, coeff_jet, (h.iteratedDeriv n).eq_of_nhds] lemma jet_const (a : ℝ) : jet (fun _ => a) = C a := by ext n by_cases hn : n = 0 <;> simp [coeff_jet, iteratedDeriv_const, coeff_C, hn] lemma jet_add {f g : ℝ → ℝ} (hf : ContDiffAt ℝ ∞ f 0) (hg : ContDiffAt ℝ ∞ g 0) : jet (fun x => f x + g x) = jet f + jet g := by ext n rw [map_add, coeff_jet, coeff_jet, coeff_jet, iteratedDeriv_fun_add (hf.of_le (show (n : ℕ∞ω) ≤ ∞ from by exact_mod_cast le_top)) (hg.of_le (show (n : ℕ∞ω) ≤ ∞ from by exact_mod_cast le_top)), add_div] lemma jet_sub {f g : ℝ → ℝ} (hf : ContDiffAt ℝ ∞ f 0) (hg : ContDiffAt ℝ ∞ g 0) : jet (fun x => f x - g x) = jet f - jet g := by ext n rw [map_sub, coeff_jet, coeff_jet, coeff_jet, iteratedDeriv_fun_sub (hf.of_le (show (n : ℕ∞ω) ≤ ∞ from by exact_mod_cast le_top)) (hg.of_le (show (n : ℕ∞ω) ≤ ∞ from by exact_mod_cast le_top)), sub_div] lemma jet_mul {f g : ℝ → ℝ} (hf : ContDiffAt ℝ ∞ f 0) (hg : ContDiffAt ℝ ∞ g 0) : jet (fun x => f x*g x) = jet f * jet g := by ext n simp only [coeff_jet, iteratedDeriv_fun_mul (hf.of_le (show (n : ℕ∞ω) ≤ ∞ from by exact_mod_cast le_top)) (hg.of_le (show (n : ℕ∞ω) ≤ ∞ from by exact_mod_cast le_top)), Finset.sum_div, PowerSeries.coeff_mul, Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk, Nat.succ_eq_add_one] refine Finset.sum_congr rfl fun i hi => ?_ rw [Nat.cast_choose _ (by have := Finset.mem_range.mp hi; omega)] field_simp [Nat.factorial_ne_zero] lemma jet_pow {f : ℝ → ℝ} (hf : ContDiffAt ℝ ∞ f 0) (n : ℕ) : jet (fun x => f x^n) = (jet f)^n := by induction n with | zero => simp only [pow_zero]; rw [jet_const, map_one] | succ n hn => simp only [pow_succ] rw [jet_mul (hf.pow n) hf, hn] lemma jet_deriv (f : ℝ → ℝ) : jet (deriv f) = derivative (jet f) := by ext n rw [coeff_derivative, coeff_jet, coeff_jet, ← iteratedDeriv_succ', Nat.factorial_succ] push_cast field_simp lemma jet_id : jet (fun x => x) = X := by ext n cases n with | zero => simp [coeff_jet] | succ n => rw [coeff_jet, iteratedDeriv_succ'] have hd : deriv (fun x : ℝ => x) = fun _ => (1 : ℝ) := by ext x; simp rw [hd, iteratedDeriv_const] cases n <;> simp [coeff_X] lemma jet_z : jet (fun x : ℝ => 1+x) = 1+X := by rw [jet_add (by fun_prop) (by fun_prop), jet_const, map_one, jet_id] lemma jet_zero (f : ℝ → ℝ) (h : f =ᶠ[𝓝 0] fun _ => 0) : jet f = 0 := by rw [jet_congr h, jet_const, map_zero] def z : Series := 1 + X def prefactor (sigma : ℝ) : Series := z * (1 - C sigma*z^2) def den (sigma : ℝ) (u : Series) : Series := prefactor sigma * (1-u^2) def r (u : Series) : Series := 1-C (3/5)*u^2 def t (kappa : ℝ) (u : Series) : Series := C kappa*z*(1-u^2)+3*(z-u) def residual (sigma kappa : ℝ) (u : Series) : Series := den sigma u * derivative u - r u * t kappa u def profileResidual (sigma kappa : ℝ) (f : ℝ → ℝ) (x : ℝ) : ℝ := (1+x)*(1-sigma*(1+x)^2)*(1-(f x)^2)*deriv f x - (1-(3/5)*(f x)^2)*(kappa*(1+x)*(1-(f x)^2)+3*((1+x)-f x)) lemma jet_residual (sigma kappa : ℝ) {f : ℝ → ℝ} (hf : ContDiffAt ℝ ∞ f 0) : jet (profileResidual sigma kappa f) = residual sigma kappa (jet f) := by have hfd : ContDiffAt ℝ ∞ (deriv f) 0 := hf.derivWithin (by simp) unfold profileResidual residual den prefactor r t z simp (disch := fun_prop) only [jet_sub, jet_mul, jet_add, jet_pow, jet_const, jet_z, jet_deriv, map_one, map_ofNat] /-- Every smooth local solution supplies an exact formal jet solution, without assuming convergence of its Taylor expansion. -/ lemma residual_of_smooth_solution (sigma kappa : ℝ) {f : ℝ → ℝ} (hf : ContDiffAt ℝ ∞ f 0) (heq : profileResidual sigma kappa f =ᶠ[𝓝 0] fun _ => 0) : residual sigma kappa (jet f) = 0 := by rw [← jet_residual sigma kappa hf] exact jet_zero _ heq end SepticProfile.SonicJet namespace SepticProfile.EulerFormal open PowerSeries Filter open scoped Topology ContDiff abbrev Series := PowerSeries ℝ def A : Series := 1-X def B (h : ℝ) : Series := 1-C (1+h)*X def pref (sigma h : ℝ) : Series := B h*(A^2-C sigma*(B h)^2) def residual (sigma kappa c h : ℝ) (u : Series) : Series := pref sigma h*(1-u^2)*derivative u+ C h*(1-C c*u^2)*(C kappa*B h*(1-u^2)+3*(B h-A*u)) def profileResidual (sigma kappa c h : ℝ) (u : ℝ → ℝ) (x : ℝ) : ℝ := (1-(1+h)*x)*((1-x)^2-sigma*(1-(1+h)*x)^2)*(1-(u x)^2)*deriv u x+ h*(1-c*(u x)^2)*(kappa*(1-(1+h)*x)*(1-(u x)^2)+3*((1-(1+h)*x)-(1-x)*u x)) lemma jet_A : SonicJet.jet (fun x : ℝ => 1-x)=A := by rw [SonicJet.jet_sub (by fun_prop) (by fun_prop),SonicJet.jet_const, SonicJet.jet_id,map_one] rfl lemma jet_B (h : ℝ) : SonicJet.jet (fun x : ℝ => 1-(1+h)*x)=B h := by rw [SonicJet.jet_sub (by fun_prop) (by fun_prop),SonicJet.jet_const, SonicJet.jet_mul (by fun_prop) (by fun_prop),SonicJet.jet_const,SonicJet.jet_id,map_one] rfl lemma jet_linear (a : ℝ) : SonicJet.jet (fun x : ℝ => a*x)=C a*X := by rw [SonicJet.jet_mul (by fun_prop) (by fun_prop),SonicJet.jet_const,SonicJet.jet_id] lemma jet_residual (sigma kappa c h : ℝ) {u : ℝ → ℝ} (hu : ContDiffAt ℝ ∞ u 0) : SonicJet.jet (profileResidual sigma kappa c h u)=residual sigma kappa c h (SonicJet.jet u) := by have hud : ContDiffAt ℝ ∞ (deriv u) 0 := hu.derivWithin (by simp) unfold profileResidual residual pref simp (disch := fun_prop) only [SonicJet.jet_add,SonicJet.jet_sub,SonicJet.jet_mul, SonicJet.jet_pow,SonicJet.jet_const,jet_linear,jet_A,SonicJet.jet_deriv,map_one,map_ofNat] rfl lemma residual_of_solution (sigma kappa c h : ℝ) {u : ℝ → ℝ} (hu : ContDiffAt ℝ ∞ u 0) (he : profileResidual sigma kappa c h u=ᶠ[𝓝 0] fun _ => 0) : residual sigma kappa c h (SonicJet.jet u)=0 := by rw [← jet_residual sigma kappa c h hu] exact SonicJet.jet_zero _ he def p0 (sigma _h : ℝ) : ℝ := 1-sigma def p1 (sigma h : ℝ) : ℝ := -2-(1+h)+3*sigma*(1+h) def p2 (sigma h : ℝ) : ℝ := 1+2*(1+h)-3*sigma*(1+h)^2 def p3 (sigma h : ℝ) : ℝ := -(1+h)+sigma*(1+h)^3 def i10 (kappa c _h : ℝ) : ℝ := -(1-c)*(2*kappa+3) def i11 (kappa c h : ℝ) : ℝ := (1-c)*(2*kappa*(1+h)+3)+6*c*h def i20 (kappa c _h : ℝ) : ℝ := kappa*(-(1-c)+4*c)+6*c def i21 (kappa c h : ℝ) : ℝ := -kappa*(-(1-c)+4*c)*(1+h)-6*c+3*c*h def i30 (kappa c _h : ℝ) : ℝ := 4*c*kappa+3*c def i31 (kappa c h : ℝ) : ℝ := -4*c*kappa*(1+h)-3*c def i40 (kappa c _h : ℝ) : ℝ := c*kappa def i41 (kappa c h : ℝ) : ℝ := -c*kappa*(1+h) lemma pref_expansion (sigma h : ℝ) : pref sigma h=C (p0 sigma h)+C (p1 sigma h)*X+C (p2 sigma h)*X^2+C (p3 sigma h)*X^3 := by unfold pref A B p0 p1 p2 p3 simp only [map_add,map_sub,map_neg,map_mul,map_pow,map_one,map_ofNat] ring /-- This expansion is what the rational interval recurrence computes, including all cubic/quartic terms and the Euler factors. -/ lemma residual_expansion (sigma kappa c h : ℝ) (w : Series) : residual sigma kappa c h (1+w) = -pref sigma h*(derivative (w^2)+C (1/3)*derivative (w^3))+ C h*(-C (3*(1-c)*h)*X+ (C (i10 kappa c h)+C (i11 kappa c h)*X)*w+ (C (i20 kappa c h)+C (i21 kappa c h)*X)*w^2+ (C (i30 kappa c h)+C (i31 kappa c h)*X)*w^3+ (C (i40 kappa c h)+C (i41 kappa c h)*X)*w^4) := by unfold residual i10 i11 i20 i21 i30 i31 i40 i41 simp only [map_add,derivative_one,zero_add,derivative_pow] norm_num only [Nat.cast_ofNat, map_mul,map_add,map_sub,map_neg,map_pow,map_one,map_ofNat] have hc : (C (1/3) : Series)*3=1 := by rw [← map_ofNat C 3,← map_mul] norm_num have h3 : ∀ v : Series, C (1/3)*(3*v)=v := by intro v; rw [← mul_assoc,hc,one_mul] simp only [mul_assoc,h3] unfold A B simp only [map_add,map_one] have hc' : (3 : Series)*C (1/3)=1 := by simpa only [mul_comm] using hc linear_combination (norm := ring_nf) -(pref sigma h*w^2*derivative w)*hc' simp only [mul_assoc,hc,mul_one,neg_add_cancel] lemma coeff_X_mul_derivative (w : Series) (n : ℕ) : coeff n (X*derivative w)=(n : ℝ)*coeff n w := by cases n with | zero => simp | succ n => rw [coeff_succ_X_mul,coeff_derivative]; push_cast; ring lemma coeff_pref_derivative (sigma h : ℝ) (w : Series) (n : ℕ) : coeff (n+2) (pref sigma h*derivative w)= p0 sigma h*(n+3)*coeff (n+3) w+ p1 sigma h*(n+2)*coeff (n+2) w+ p2 sigma h*(n+1)*coeff (n+1) w+ p3 sigma h*n*coeff n w := by rw [pref_expansion] simp only [add_mul,map_add] have h0 : coeff (n+2) (C (p0 sigma h)*derivative w)=p0 sigma h*(n+3)*coeff (n+3) w := by rw [coeff_C_mul,coeff_derivative] push_cast ring have h1 : coeff (n+2) (C (p1 sigma h)*X*derivative w)=p1 sigma h*(n+2)*coeff (n+2) w := by rw [mul_assoc,coeff_C_mul,coeff_X_mul_derivative] push_cast ring have h2 : coeff (n+2) (C (p2 sigma h)*X^2*derivative w)=p2 sigma h*(n+1)*coeff (n+1) w := by rw [mul_assoc,coeff_C_mul,show X^2*derivative w=X*(X*derivative w) by ring, show n+2=(n+1)+1 by omega,coeff_succ_X_mul,coeff_X_mul_derivative] push_cast ring have h3 : coeff (n+2) (C (p3 sigma h)*X^3*derivative w)=p3 sigma h*n*coeff n w := by rw [mul_assoc,coeff_C_mul,show X^3*derivative w=X^2*(X*derivative w) by ring, coeff_X_pow_mul,coeff_X_mul_derivative] ring rw [h0,h1,h2,h3] lemma coeff_affine_mul (a b : ℝ) (w : Series) (n : ℕ) : coeff (n+1) ((C a+C b*X)*w)=a*coeff (n+1) w+b*coeff n w := by rw [add_mul,map_add,coeff_C_mul,mul_assoc,coeff_C_mul,coeff_succ_X_mul] lemma coefficient_equation (sigma kappa c h : ℝ) (w : Series) (n : ℕ) (he : residual sigma kappa c h (1+w)=0) : -(p0 sigma h*(n+3)*(coeff (n+3) (w^2)+coeff (n+3) (w^3)/3)+ p1 sigma h*(n+2)*(coeff (n+2) (w^2)+coeff (n+2) (w^3)/3)+ p2 sigma h*(n+1)*(coeff (n+1) (w^2)+coeff (n+1) (w^3)/3)+ p3 sigma h*n*(coeff n (w^2)+coeff n (w^3)/3))+ h*(i10 kappa c h*coeff (n+2) w+i11 kappa c h*coeff (n+1) w+ i20 kappa c h*coeff (n+2) (w^2)+i21 kappa c h*coeff (n+1) (w^2)+ i30 kappa c h*coeff (n+2) (w^3)+i31 kappa c h*coeff (n+1) (w^3)+ i40 kappa c h*coeff (n+2) (w^4)+i41 kappa c h*coeff (n+1) (w^4))=0 := by have he' := congrArg (coeff (n+2) : Series → ℝ) he rw [residual_expansion] at he' have hs : -pref sigma h*(derivative (w^2)+C (1/3)*derivative (w^3)) = -(pref sigma h*derivative (w^2))-C (1/3)*(pref sigma h*derivative (w^3)) := by ring rw [hs] at he' simp only [map_add,map_sub,map_neg,map_zero,coeff_C_mul,coeff_pref_derivative, coeff_affine_mul] at he' have hx : coeff (n+2) (-C (3*(1-c)*h)*X)=0 := by rw [neg_mul,map_neg,coeff_C_mul] simp [coeff_X,show n+2≠1 by omega] rw [hx] at he' linear_combination he' end SepticProfile.EulerFormal namespace SepticProfile.EulerFormal open PowerSeries Finset lemma coeff_mul_Ico (f g : Series) (a b n : ℕ) (hf : ∀ i nlinarith lemma rounding_bound (a b : ℤ) (hb : 0 < b) : |(a : ℝ)-(b : ℝ)*(a/b : ℤ)| ≤ (b : ℝ) := by obtain ⟨hl,hu⟩ := quotient_bounds a b hb have hl' : ((a/b : ℤ) : ℝ)*(b : ℝ) ≤ (a : ℝ) := by exact_mod_cast hl have hu' : (a : ℝ) < (((a/b : ℤ) : ℝ)+1)*(b : ℝ) := by exact_mod_cast hu rw [abs_of_nonneg (by nlinarith)] nlinarith lemma holds_zero (Q : ℤ) : Holds Q zero 0 := by simp [Holds,zero] lemma holds_neg {Q : ℤ} {b : Box} {x : ℝ} (hx : Holds Q b x) : Holds Q (neg b) (-x) := by unfold Holds neg push_cast calc _ = |(-((Q : ℝ)*x-b.center))| := by congr 1; ring _ = |(Q : ℝ)*x-b.center| := abs_neg _ _ ≤ _ := hx lemma holds_add {Q : ℤ} {a b : Box} {x y : ℝ} (hx : Holds Q a x) (hy : Holds Q b y) : Holds Q (add a b) (x+y) := by change |(Q : ℝ)*(x+y)-((a.center+b.center : ℤ) : ℝ)| ≤ ((a.radius+b.radius : ℤ) : ℝ) push_cast calc _ = |((Q : ℝ)*x-a.center)+((Q : ℝ)*y-b.center)| := by congr 1; ring _ ≤ |(Q : ℝ)*x-a.center| +|(Q : ℝ)*y-b.center| := abs_add_le _ _ _ ≤ _ := add_le_add hx hy lemma holds_sub {Q : ℤ} {a b : Box} {x y : ℝ} (hx : Holds Q a x) (hy : Holds Q b y) : Holds Q (sub a b) (x-y) := by simpa only [sub, sub_eq_add_neg] using holds_add hx (holds_neg hy) lemma holds_scale {Q a b : ℤ} {v : Box} {x : ℝ} (hb : 0 < b) (hx : Holds Q v x) : Holds Q (scale a b v) ((a : ℝ)/(b : ℝ)*x) := by have hb' : 0 < (b : ℝ) := by exact_mod_cast hb have hr := radius_nonneg hx have hround := rounding_bound (a*v.center) b hb have hrad := (quotient_bounds (|a| *v.radius) b hb).2 have hrad' : (|a| : ℝ)*(v.radius : ℝ) < (((|a| *v.radius)/b : ℤ)+1 : ℝ)*(b : ℝ) := by exact_mod_cast hrad unfold Holds scale push_cast apply (mul_le_mul_iff_of_pos_left hb').mp rw [← abs_of_pos hb',← abs_mul] simp only [abs_of_pos hb'] calc _ = |(a : ℝ)*((Q : ℝ)*x-v.center)+ ((a*v.center : ℤ) : ℝ)-(b : ℝ)*((a*v.center/b : ℤ) : ℝ)| := by push_cast congr 1 (field_simp; ring) _ ≤ |(a : ℝ)*((Q : ℝ)*x-v.center)| + |((a*v.center : ℤ) : ℝ)-(b : ℝ)*((a*v.center/b : ℤ) : ℝ)| := by convert abs_add_le ((a : ℝ)*((Q : ℝ)*x-v.center)) (((a*v.center : ℤ) : ℝ)-(b : ℝ)*((a*v.center/b : ℤ) : ℝ)) using 1; congr 1; ring _ ≤ |(a : ℝ)| *(v.radius : ℝ)+(b : ℝ) := by rw [abs_mul] exact add_le_add (mul_le_mul_of_nonneg_left hx (abs_nonneg _)) hround _ ≤ _ := by rw [← Int.cast_abs] at hrad' ⊢ nlinarith lemma holds_mul {Q : ℤ} {a b : Box} {x y : ℝ} (hQ : 0 < Q) (hx : Holds Q a x) (hy : Holds Q b y) : Holds Q (mul Q a b) (x*y) := by have hQ' : 0 < (Q : ℝ) := by exact_mod_cast hQ have ha := radius_nonneg hx have hb := radius_nonneg hy have har : (0 : ℝ) ≤ a.radius := by exact_mod_cast ha have hbr : (0 : ℝ) ≤ b.radius := by exact_mod_cast hb have hround := rounding_bound (a.center*b.center) Q hQ let E : ℤ := |a.center| *b.radius+|b.center| *a.radius+a.radius*b.radius have hrad := (quotient_bounds E Q hQ).2 have hrad' : (E : ℝ) < (((E/Q : ℤ) : ℝ)+1)*(Q : ℝ) := by exact_mod_cast hrad have he : |(Q : ℝ)^2*(x*y)-(a.center : ℝ)*b.center| ≤ (E : ℝ) := by have hid : (Q : ℝ)^2*(x*y)-(a.center : ℝ)*b.center = (a.center : ℝ)*((Q : ℝ)*y-b.center) + (b.center : ℝ)*((Q : ℝ)*x-a.center) + ((Q : ℝ)*x-a.center)*((Q : ℝ)*y-b.center) := by ring rw [hid] calc _ ≤ |(a.center : ℝ)*((Q : ℝ)*y-b.center)| + |(b.center : ℝ)*((Q : ℝ)*x-a.center)| + |((Q : ℝ)*x-a.center)*((Q : ℝ)*y-b.center)| := (abs_add_le _ _).trans (add_le_add (abs_add_le _ _) le_rfl) _ ≤ |(a.center : ℝ)| *b.radius+|(b.center : ℝ)| *a.radius+ (a.radius : ℝ)*b.radius := by simp only [abs_mul] exact add_le_add (add_le_add (mul_le_mul_of_nonneg_left hy (abs_nonneg _)) (mul_le_mul_of_nonneg_left hx (abs_nonneg _))) (mul_le_mul hx hy (abs_nonneg _) har) _ = _ := by simp [E] unfold Holds mul push_cast apply (mul_le_mul_iff_of_pos_left hQ').mp rw [← abs_of_pos hQ',← abs_mul] simp only [abs_of_pos hQ'] calc _ = |((Q : ℝ)^2*(x*y)-(a.center : ℝ)*b.center)+ ((a.center*b.center : ℤ) : ℝ)-(Q : ℝ)*((a.center*b.center/Q : ℤ) : ℝ)| := by push_cast congr 1 ring _ ≤ |(Q : ℝ)^2*(x*y)-(a.center : ℝ)*b.center| + |((a.center*b.center : ℤ) : ℝ)-(Q : ℝ)*((a.center*b.center/Q : ℤ) : ℝ)| := by convert abs_add_le ((Q : ℝ)^2*(x*y)-(a.center : ℝ)*b.center) (((a.center*b.center : ℤ) : ℝ)-(Q : ℝ)*((a.center*b.center/Q : ℤ) : ℝ)) using 1; congr 1; ring _ ≤ (E : ℝ)+(Q : ℝ) := add_le_add he hround _ ≤ _ := by change (E : ℝ)+(Q : ℝ) ≤ (Q : ℝ)*(((E/Q : ℤ) : ℝ)+2) nlinarith lemma holds_ofRat {Q a b : ℤ} (_hQ : 0 < Q) (hb : 0 < b) : Holds Q (ofRat Q a b) ((a : ℝ)/(b : ℝ)) := by have hb' : 0 < (b : ℝ) := by exact_mod_cast hb have hr := rounding_bound (Q*a) b hb unfold Holds ofRat push_cast apply (mul_le_mul_iff_of_pos_left hb').mp rw [← abs_of_pos hb',← abs_mul] simp only [abs_of_pos hb'] have hh : (b : ℝ)*((Q : ℝ)*((a : ℝ)/(b : ℝ))-((Q*a/b : ℤ) : ℝ)) = ((Q*a : ℤ) : ℝ)-(b : ℝ)*((Q*a/b : ℤ) : ℝ) := by push_cast field_simp rw [hh] simpa using hr end SepticProfile.FixedInterval namespace SepticProfile.FixedInterval open Finset def treeSumBox : ℕ → (ℕ → Box) → ℕ → Box | 0, f, a => f a | d+1, f, a => add (treeSumBox d f a) (treeSumBox d f (a+2^d)) lemma holds_treeSumBox {Q : ℤ} (d : ℕ) (f : ℕ → Box) (g : ℕ → ℝ) (a : ℕ) (hf : ∀ i<2^d, Holds Q (f (a+i)) (g (a+i))) : Holds Q (treeSumBox d f a) (∑ i ∈ range (2^d), g (a+i)) := by induction d generalizing a with | zero => simpa [treeSumBox] using hf 0 (by simp) | succ d ih => rw [treeSumBox,pow_succ,Nat.mul_two,Finset.sum_range_add] have h1 := ih a (fun i hi => hf i (by rw [pow_succ,Nat.mul_two]; omega)) have h2 := ih (a+2^d) (fun i hi => by simpa only [Nat.add_assoc] using hf (2^d+i) (by rw [pow_succ,Nat.mul_two]; omega)) convert holds_add h1 h2 using 1 congr 1 apply Finset.sum_congr rfl intro i hi congr 1 omega def sumBox (d lo hi : ℕ) (f : ℕ → Box) : Box := treeSumBox d (fun i => if lo ≤ i ∧ i < hi then f i else zero) 0 lemma holds_sumBox {Q : ℤ} (d lo hi : ℕ) (f : ℕ → Box) (g : ℕ → ℝ) (hbound : hi ≤ 2^d) (hf : ∀ i, lo ≤ i → i < hi → Holds Q (f i) (g i)) : Holds Q (sumBox d lo hi f) (∑ i ∈ Ico lo hi, g i) := by have hs := holds_treeSumBox d (fun i => if lo ≤ i ∧ i < hi then f i else zero) (fun i => if lo ≤ i ∧ i < hi then g i else 0) 0 (by intro i hi' simp only [zero_add] split_ifs with h · exact hf i h.1 h.2 · exact holds_zero Q) change Holds Q (sumBox d lo hi f) _ at hs convert hs using 1 simp only [zero_add] symm rw [← Finset.sum_filter] congr 1 ext i simp only [mem_filter,mem_range,mem_Ico] omega def convBox (Q : ℤ) (d lo hi : ℕ) (a b : ℕ → Box) (n : ℕ) : Box := sumBox d lo hi (fun i => mul Q (a i) (b (n-i))) lemma holds_convBox {Q : ℤ} (hQ : 0 < Q) (d lo hi : ℕ) (a b : ℕ → Box) (f g : ℕ → ℝ) (n : ℕ) (hbound : hi ≤ 2^d) (hf : ∀ i, lo ≤ i → i < hi → Holds Q (a i) (f i)) (hg : ∀ i, lo ≤ i → i < hi → Holds Q (b (n-i)) (g (n-i))) : Holds Q (convBox Q d lo hi a b n) (∑ i ∈ Ico lo hi, f i*g (n-i)) := by apply holds_sumBox d lo hi _ _ hbound intro i hi hi' exact holds_mul hQ (hf i hi hi') (hg i hi hi') lemma holds_exact (Q a : ℤ) : Holds Q ⟨Q*a,0⟩ (a : ℝ) := by simp [Holds] lemma holds_enlarge {Q : ℤ} {a b : Box} {x : ℝ} (hab : |a.center-b.center|+a.radius ≤ b.radius) (hx : Holds Q a x) : Holds Q b x := by have hab' : |(a.center : ℝ)-b.center|+(a.radius : ℝ) ≤ b.radius := by exact_mod_cast hab calc _ = |((Q : ℝ)*x-a.center)+((a.center : ℝ)-b.center)| := by congr 1; ring _ ≤ |(Q : ℝ)*x-a.center|+|(a.center : ℝ)-b.center| := abs_add_le _ _ _ ≤ _ := by unfold Holds at hx; linarith end SepticProfile.FixedInterval namespace SepticProfile.FixedInterval def rscale (a : ℚ) (v : Box) : Box := scale a.num a.den v def ratBox (Q : ℤ) (a : ℚ) : Box := ofRat Q a.num a.den lemma holds_rscale {Q : ℤ} (a : ℚ) {v : Box} {x : ℝ} (hx : Holds Q v x) : Holds Q (rscale a v) ((a : ℝ)*x) := by have hd : (0 : ℤ) intro hk by_cases hk0 : k=0 · subst k rw [hw0,ht.initial0] exact holds_zero Q by_cases hk1 : k=1 · subst k rw [hw1,ht.initial1] exact holds_ratBox hQ p.a have hk2 : 2≤k := by omega have hbk : ∀ j≤k, Holds Q (t.b j) (b j) := by intro j hj rw [ht.square j (by omega),hb j] apply holds_convBox hQ D 1 j _ _ w w j (by omega) · intro i hi hi' exact ih i (by omega) (by omega) · intro i hi hi' exact ih (j-i) (by omega) (by omega) have hck : ∀ j≤k+1, Holds Q (t.c j) (c j) := by intro j hj rw [ht.cube j (by omega),hc j] apply holds_convBox hQ D 1 (j-1) _ _ w b j (by omega) · intro i hi hi' exact ih i (by omega) (by omega) · intro i hi hi' exact hbk (j-i) (by omega) have hdk : ∀ j≤k, Holds Q (t.d j) (d j) := by intro j hj rw [ht.fourth j (by omega),hd j] apply holds_convBox hQ D 2 (j-1) _ _ b b j (by omega) · intro i hi hi' exact hbk i (by omega) · intro i hi hi' exact hbk (j-i) (by omega) have hbrk : Holds Q (t.br k) (br k) := by rw [ht.squareR k hk2 hk,hbr k] apply holds_convBox hQ D 2 k _ _ w w (k+1) (by omega) · intro i hi hi' exact ih i (by omega) (by omega) · intro i hi hi' exact ih (k+1-i) (by omega) (by omega) have hr := holds_stepBox p t w b c d br k (ih (k-1) (by omega) (by omega)) hbk hck hdk hbrk have he' := he k hk2 hk have hn : (linear p k : ℝ)≠0 := by exact_mod_cast ht.nonzero k hk2 hk have hf : w k=((-1/linear p k : ℚ) : ℝ)*stepValue p w b c d br k := by push_cast field_simp linear_combination he' rw [hf,ht.update k hk2 hk,ht.residue k hk2 hk] exact holds_rscale _ hr end SepticProfile.JetTrace namespace SepticProfile.JetTrace open PowerSeries Finset def sourceParams (sigma kappa c h a : ℚ) : Parameters where p0 := 1-sigma p1 := -2-(1+h)+3*sigma*(1+h) p2 := 1+2*(1+h)-3*sigma*(1+h)^2 p3 := -(1+h)+sigma*(1+h)^3 i10 := -(1-c)*(2*kappa+3) i11 := (1-c)*(2*kappa*(1+h)+3)+6*c*h i20 := kappa*(-(1-c)+4*c)+6*c i21 := -kappa*(-(1-c)+4*c)*(1+h)-6*c+3*c*h i30 := 4*c*kappa+3*c i31 := -4*c*kappa*(1+h)-3*c i40 := c*kappa i41 := -c*kappa*(1+h) h := h a := a lemma source_recurrence (sigma kappa c h a : ℚ) (w : EulerFormal.Series) (hw0 : coeff 0 w=0) (hw1 : coeff 1 w=(a : ℝ)) (he : EulerFormal.residual sigma kappa c h (1+w)=0) (k : ℕ) (hk : 2≤k) : (linear (sourceParams sigma kappa c h a) k : ℝ)*coeff k w+ stepValue (sourceParams sigma kappa c h a) (fun i => coeff i w) (fun i => coeff i (w^2)) (fun i => coeff i (w^3)) (fun i => coeff i (w^4)) (fun j => ∑ i ∈ Ico 2 j, coeff i w*coeff (j+1-i) w) k=0 := by obtain ⟨n,rfl⟩ : ∃ n, k=n+2 := ⟨k-2,by omega⟩ have hh := EulerFormal.coefficient_equation (sigma : ℝ) kappa c h w n he have hs := EulerFormal.coeff_square_remainder w hw0 (n+2) (by omega) rw [hw1] at hs rw [hs] at hh unfold linear stepValue sourceParams simp only [show n+2-1=n+1 by omega,Nat.add_sub_cancel_right] simp only [EulerFormal.p0,EulerFormal.p1,EulerFormal.p2,EulerFormal.p3, EulerFormal.i10,EulerFormal.i11,EulerFormal.i20,EulerFormal.i21,EulerFormal.i30, EulerFormal.i31,EulerFormal.i40,EulerFormal.i41] at hh push_cast linear_combination hh theorem source_coefficients_enclosed {Q : ℤ} (hQ : 0 coeff i w) (fun i => coeff i (w^2)) (fun i => coeff i (w^3)) (fun i => coeff i (w^4)) (fun j => ∑ i ∈ Ico 2 j, coeff i w*coeff (j+1-i) w) hw0 hw1 · exact EulerFormal.coeff_square w hw0 · exact EulerFormal.coeff_cube w hw0 · exact EulerFormal.coeff_fourth w hw0 · intro j rfl · intro j hj hj' exact source_recurrence sigma kappa c h a w hw0 hw1 he j hj end SepticProfile.JetTrace namespace SepticProfile.EulerFormal open PowerSeries lemma residual_direct (sigma kappa c h : ℝ) (u : Series) : residual sigma kappa c h u = pref sigma h*(derivative u-C (1/3)*derivative (u^3))+ C h*(C (kappa+3)*B h-3*A*u-C (kappa+c*(kappa+3))*B h*u^2+ C (3*c)*A*u^3+C (c*kappa)*B h*u^4) := by unfold residual simp only [derivative_pow] norm_num only [Nat.cast_ofNat,map_add,map_mul,map_ofNat] have hc : (C (1/3) : Series)*3=1 := by rw [← map_ofNat C 3,← map_mul] norm_num have h3 : ∀ v : Series, C (1/3)*(3*v)=v := by intro v; rw [← mul_assoc,hc,one_mul] simp only [mul_assoc,h3] ring lemma coeff_B_mul (h : ℝ) (u : Series) (n : ℕ) : coeff (n+1) (B h*u)=coeff (n+1) u-(1+h)*coeff n u := by simp only [B,sub_mul,one_mul,mul_assoc,map_sub (coeff (n+1)),coeff_C_mul,coeff_succ_X_mul] lemma coeff_A_mul (u : Series) (n : ℕ) : coeff (n+1) (A*u)=coeff (n+1) u-coeff n u := by simp [A,sub_mul,coeff_succ_X_mul] lemma coeff_B_high (h : ℝ) (n : ℕ) : coeff (n+2) (B h)=0 := by simp [B] lemma residual_coeff_direct (sigma kappa c h : ℝ) (u : Series) (n : ℕ) : coeff (n+2) (residual sigma kappa c h u)= p0 sigma h*(n+3)*(coeff (n+3) u-coeff (n+3) (u^3)/3)+ p1 sigma h*(n+2)*(coeff (n+2) u-coeff (n+2) (u^3)/3)+ p2 sigma h*(n+1)*(coeff (n+1) u-coeff (n+1) (u^3)/3)+ p3 sigma h*n*(coeff n u-coeff n (u^3)/3)+ h*(-3*coeff (n+2) u+3*coeff (n+1) u- (kappa+c*(kappa+3))*coeff (n+2) (u^2)+ (kappa+c*(kappa+3))*(1+h)*coeff (n+1) (u^2)+ 3*c*coeff (n+2) (u^3)-3*c*coeff (n+1) (u^3)+ c*kappa*coeff (n+2) (u^4)-c*kappa*(1+h)*coeff (n+1) (u^4)) := by rw [residual_direct] have hx : pref sigma h*(derivative u-C (1/3)*derivative (u^3))= pref sigma h*derivative u-C (1/3)*(pref sigma h*derivative (u^3)) := by ring rw [hx] have h2 : 3*A*u=(C 3)*(A*u) := by rw [map_ofNat]; ring rw [h2] simp only [mul_assoc,coeff_C_mul,map_add (coeff (n+2)),map_sub (coeff (n+2)),coeff_pref_derivative, coeff_B_mul,coeff_A_mul,coeff_B_high,mul_zero] ring /-- Tangency of the sonic vector-field divisor prevents loss of one order in its residual. This is the reason finite jets suffice for the barrier. -/ lemma residual_congr_X_pow (sigma kappa c h : ℝ) (u v : Series) (m : ℕ) (hm : 1 ≤ m) (hu0 : coeff 0 u=1) (hdiv : X^m ∣ u-v) : X^m ∣ residual sigma kappa c h u-residual sigma kappa c h v := by obtain ⟨w,hw⟩ := hdiv have hu : u=v+X^m*w := by linear_combination hw have hu2 : X^m ∣ u^2-v^2 := by refine ⟨w*(u+v),?_⟩ linear_combination (u+v)*hw have hx : X ∣ 1-u^2 := by rw [X_dvd_iff] have hu0' : constantCoeff u=1 := by simpa using hu0 simp only [map_sub,map_one,map_pow,hu0',one_pow,sub_self] obtain ⟨z,hz⟩ := hx have hd : derivative u-derivative v= (m : ℝ) • (X^(m-1)*w)+X^m*derivative w := by rw [hu,map_add,add_sub_cancel_left,Derivation.leibniz,derivative_pow,derivative_X] simp only [mul_one] simp only [Algebra.smul_def,Algebra.algebraMap_self_apply] push_cast ring have hd1 : X^m ∣ (1-u^2)*(derivative u-derivative v) := by refine ⟨(m : ℝ) • (z*w)+X*z*derivative w,?_⟩ rw [hd,hz] simp only [Algebra.smul_def] have hp : X*X^(m-1)=(X : Series)^m := by rw [← pow_succ']; congr 1; omega calc _ = (m : ℝ) • ((X*X^(m-1))*(z*w))+X^m*(X*z*derivative w) := by simp only [Algebra.smul_def]; ring _ = _ := by rw [hp]; simp only [Algebra.smul_def]; ring have hu2' : X^m ∣ (1-u^2)-(1-v^2) := by (convert hu2.neg_right using 1; ring) have hres1 : X^m ∣ (1-u^2)*derivative u-(1-v^2)*derivative v := by have := hd1.add (hu2'.mul_right (derivative v)) (convert this using 1; ring) have hbr : X^m ∣ (C kappa*B h*(1-u^2)+3*(B h-A*u))- (C kappa*B h*(1-v^2)+3*(B h-A*v)) := by have h12 := hu2'.mul_left (C kappa*B h) have h13 : X^m ∣ -3*A*(u-v) := (show X^m ∣ u-v from ⟨w,hw⟩).mul_left (-3*A) (convert h12.add h13 using 1; ring) have hup : X^m ∣ (1-C c*u^2)-(1-C c*v^2) := by (convert hu2.mul_left (-C c) using 1; ring) have hres2 := (hbr.mul_left (1-C c*u^2)).add (hup.mul_right (C kappa*B h*(1-v^2)+3*(B h-A*v))) have hh := (hres1.mul_left (pref sigma h)).add (hres2.mul_left (C h)) (convert hh using 1; unfold residual; ring) end SepticProfile.EulerFormal namespace SepticProfile.FixedInterval open PowerSeries Finset lemma coeff_mul_support (f g : PowerSeries ℝ) (A B n : ℕ) (hf : ∀ i, A ≤ i → coeff i f=0) (hg : ∀ i, B ≤ i → coeff i g=0) : coeff n (f*g)=∑ i ∈ Ico (n+1-B) (min A (n+1)), coeff i f*coeff (n-i) g := by rw [coeff_mul,Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk] symm apply Finset.sum_subset · intro i hi simp only [mem_Ico,mem_range] at hi ⊢ omega · intro i hi hnot have hi' := mem_range.mp hi by_cases ha : A ≤ i · rw [hf i ha,zero_mul] · have hb : B ≤ n-i := by simp only [mem_Ico,not_and] at hnot omega rw [hg (n-i) hb,mul_zero] lemma coeff_mul_support_zero (f g : PowerSeries ℝ) (A B n : ℕ) (hf : ∀ i, A ≤ i → coeff i f=0) (hg : ∀ i, B ≤ i → coeff i g=0) (hn : A+B-1 ≤ n) : coeff n (f*g)=0 := by rw [coeff_mul_support f g A B n hf hg] have : Ico (n+1-B) (min A (n+1))=∅ := by apply Ico_eq_empty_of_le; omega rw [this,sum_empty] lemma holds_poly_conv {Q : ℤ} (hQ : 0 coeff i f) (fun i => coeff i g) n · exact le_trans (min_le_left _ _) hA · intro i _ _; exact ha i · intro i _ _; exact hb _ end SepticProfile.FixedInterval namespace SepticProfile.BarrierTrace open FixedInterval Finset structure Parameters where (p0 p1 p2 p3 b20 b21 b30 b31 b40 b41 h : ℚ) def sourceParams (sigma kappa c h : ℚ) : Parameters where p0 := 1-sigma p1 := -2-(1+h)+3*sigma*(1+h) p2 := 1+2*(1+h)-3*sigma*(1+h)^2 p3 := -(1+h)+sigma*(1+h)^3 b20 := -(kappa+c*(kappa+3)) b21 := (kappa+c*(kappa+3))*(1+h) b30 := 3*c b31 := -3*c b40 := c*kappa b41 := -c*kappa*(1+h) h := h structure Trace where (u b c d r : ℕ → Box) def residualBox (p : Parameters) (t : Trace) (k : ℕ) : Box := add (add (add (add (rscale (p.p0*(k+1)) (sub (t.u (k+1)) (rscale (1/3) (t.c (k+1))))) (rscale (p.p1*k) (sub (t.u k) (rscale (1/3) (t.c k))))) (rscale (p.p2*(k-1)) (sub (t.u (k-1)) (rscale (1/3) (t.c (k-1)))))) (rscale (p.p3*(k-2)) (sub (t.u (k-2)) (rscale (1/3) (t.c (k-2)))))) (rscale p.h (add (add (add (add (add (add (add (rscale (-3) (t.u k)) (rscale 3 (t.u (k-1)))) (rscale p.b20 (t.b k))) (rscale p.b21 (t.b (k-1)))) (rscale p.b30 (t.c k))) (rscale p.b31 (t.c (k-1)))) (rscale p.b40 (t.d k))) (rscale p.b41 (t.d (k-1))))) lemma residual_enclosed {Q : ℤ} (sigma kappa c h : ℚ) (t : Trace) (u : EulerFormal.Series) (k : ℕ) (hk : 2 ≤ k) (hu : ∀ i, Holds Q (t.u i) (PowerSeries.coeff i u)) (hb : ∀ i, Holds Q (t.b i) (PowerSeries.coeff i (u^2))) (hc : ∀ i, Holds Q (t.c i) (PowerSeries.coeff i (u^3))) (hd : ∀ i, Holds Q (t.d i) (PowerSeries.coeff i (u^4))) : Holds Q (residualBox (sourceParams sigma kappa c h) t k) (PowerSeries.coeff k (EulerFormal.residual sigma kappa c h u)) := by obtain ⟨n,rfl⟩ : ∃ n, k=n+2 := ⟨k-2,by omega⟩ rw [EulerFormal.residual_coeff_direct] unfold residualBox sourceParams simp only [show n+2-1=n+1 by omega,Nat.add_sub_cancel_right] have hs : ∀ b : Box, ∀ x : ℝ, ∀ r : ℚ, Holds Q b x → Holds Q (rscale r b) ((r : ℝ)*x) := fun _ _ _ hx => holds_rscale _ hx have hbound := holds_add (holds_add (holds_add (holds_add (hs _ _ ((1-sigma)*(n+2+1)) (holds_sub (hu (n+2+1)) (hs _ _ (1/3) (hc (n+2+1))))) (hs _ _ ((-2-(1+h)+3*sigma*(1+h))*(n+2)) (holds_sub (hu (n+2)) (hs _ _ (1/3) (hc (n+2)))))) (hs _ _ ((1+2*(1+h)-3*sigma*(1+h)^2)*(n+1)) (holds_sub (hu (n+1)) (hs _ _ (1/3) (hc (n+1)))))) (hs _ _ ((-(1+h)+sigma*(1+h)^3)*n) (holds_sub (hu n) (hs _ _ (1/3) (hc n)))) ) (hs _ _ h (holds_add (holds_add (holds_add (holds_add (holds_add (holds_add (holds_add (hs _ _ (-3) (hu (n+2))) (hs _ _ 3 (hu (n+1)))) (hs _ _ (-(kappa+c*(kappa+3))) (hb (n+2)))) (hs _ _ ((kappa+c*(kappa+3))*(1+h)) (hb (n+1)))) (hs _ _ (3*c) (hc (n+2)))) (hs _ _ (-3*c) (hc (n+1)))) (hs _ _ (c*kappa) (hd (n+2)))) (hs _ _ (-c*kappa*(1+h)) (hd (n+1))))) convert hbound using 1 · push_cast (congr 1; congr 1; congr 1) all_goals (congr 1; ring) · unfold EulerFormal.p0 EulerFormal.p1 EulerFormal.p2 EulerFormal.p3 push_cast ring end SepticProfile.BarrierTrace namespace SepticProfile.FixedInterval lemma holds_change_scale {Q S : ℤ} (hS : 0 linarith lemma abs_bound {Q : ℤ} {b : Box} {x : ℝ} (hx : Holds Q b x) : |(Q : ℝ)*x| ≤ (|b.center|+b.radius : ℤ) := by calc _ = |((Q : ℝ)*x-b.center)+(b.center : ℝ)| := by congr 1; ring _ ≤ |(Q : ℝ)*x-b.center|+|(b.center : ℝ)| := abs_add_le _ _ _ ≤ _ := by norm_cast; simpa [add_comm] using add_le_add_right hx |(b.center : ℝ)| end SepticProfile.FixedInterval namespace SepticProfile.BarrierTrace open FixedInterval PowerSeries structure ValidPowers (Q : ℤ) (D N : ℕ) (t : Trace) : Prop where square : ∀ n, n≤2*N → t.b n=convBox Q D (n-N) (min (N+1) (n+1)) t.u t.u n cube : ∀ n, n≤3*N → t.c n=convBox Q D (n-2*N) (min (N+1) (n+1)) t.u t.b n fourth : ∀ n, n≤4*N → t.d n=convBox Q D (n-2*N) (min (2*N+1) (n+1)) t.b t.b n bZero : ∀ n, 2*N p.eval t) x := by simp [residual,EulerFormal.profileResidual,Polynomial.deriv] lemma degree_residual (sigma kappa c h : ℝ) (p : ℝ[X]) (hp : p.natDegree≤220) : (residual sigma kappa c h p).natDegree≤881 := by have hd := natDegree_derivative_le p unfold residual compute_degree! omega noncomputable def barrier (f : PowerSeries ℝ) (δ : ℝ) : ℝ[X] := PowerSeries.trunc 221 f+C δ*X^140 lemma coeff_barrier (f : PowerSeries ℝ) (δ : ℝ) (n : ℕ) : (barrier f δ).coeff n=(if n<221 then PowerSeries.coeff n f else 0)+ (if n=140 then δ else 0) := by unfold barrier rw [coeff_add,PowerSeries.coeff_trunc,coeff_C_mul,coeff_X_pow] simp only [mul_ite,mul_one,mul_zero] lemma degree_barrier (f : PowerSeries ℝ) (δ : ℝ) : (barrier f δ).natDegree≤220 := by have hp := PowerSeries.natDegree_trunc_lt f 220 unfold barrier apply natDegree_add_le_of_degree_le · exact Nat.le_of_lt_succ hp · exact (natDegree_C_mul_X_pow_le δ 140).trans (by norm_num) lemma barrier_congr (f : PowerSeries ℝ) (δ : ℝ) : (PowerSeries.X : PowerSeries ℝ)^140 ∣ f-(barrier f δ : PowerSeries ℝ) := by rw [PowerSeries.X_pow_dvd_iff] intro n hn rw [map_sub,Polynomial.coeff_coe,coeff_barrier] simp [show n<221 by omega,show n≠140 by omega] lemma residual_lowzero (sigma kappa c h : ℝ) (f : PowerSeries ℝ) (δ : ℝ) (hf0 : PowerSeries.coeff 0 f=1) (hf : EulerFormal.residual sigma kappa c h f=0) (n : ℕ) (hn : n<140) : (residual sigma kappa c h (barrier f δ)).coeff n=0 := by have hh := EulerFormal.residual_congr_X_pow sigma kappa c h f (barrier f δ) 140 (by norm_num) hf0 (barrier_congr f δ) rw [hf,zero_sub,← coe_residual] at hh have hh' := PowerSeries.X_pow_dvd_iff.mp hh n hn simpa using hh' lemma eval_factor (p : ℝ[X]) (hp : p.natDegree≤881) (hz : ∀ n<140, p.coeff n=0) (x : ℝ) : p.eval x=x^140*∑ i ∈ Finset.range 742, p.coeff (i+140)*x^i := by rw [eval_eq_sum_range' (show p.natDegree<882 by omega), show 882=140+742 by norm_num,Finset.sum_range_add] have hzero : ∑ i ∈ Finset.range 140, p.coeff i*x^i=0 := by apply Finset.sum_eq_zero intro i hi rw [hz i (Finset.mem_range.mp hi),zero_mul] rw [hzero,zero_add,Finset.mul_sum] apply Finset.sum_congr rfl intro i hi rw [pow_add,Nat.add_comm 140 i] ring end SepticProfile.EulerPolynomial namespace SepticProfile.FixedInterval open Finset def treeInt : ℕ → (ℕ → ℤ) → ℕ → ℤ | 0,f,a => f a | D+1,f,a => treeInt D f a+treeInt D f (a+2^D) lemma treeInt_eq (D : ℕ) (f : ℕ → ℤ) (a : ℕ) : treeInt D f a=∑ i ∈ range (2^D), f (a+i) := by induction D generalizing a with | zero => simp [treeInt] | succ D ih => rw [treeInt,pow_succ,Nat.mul_two,Finset.sum_range_add,ih,ih] congr 1 apply Finset.sum_congr rfl intro i hi congr 1 omega def sumInt (D lo hi : ℕ) (f : ℕ → ℤ) : ℤ := treeInt D (fun i => if lo ≤ i ∧ i (|(b i).center|+(b i).radius)*5^i*6^(hi-1-i)) lemma scaled_tail_bound {Q : ℤ} (hQ : 0 a i*x^i) (by norm_num : 0 ≤ 3) (by norm_num : 3 ≤ 10) have hs' := Finset.sum_Ico_consecutive (fun i => a i*x^i) (by norm_num : 0 ≤ 10) hM calc 0 < (∑ i ∈ Ico 0 3, a i*x^i)+(∑ i ∈ Ico 3 10, a i*x^i)+(∑ i ∈ Ico 10 M, a i*x^i) := by linarith _ = _ := by rw [hs,hs',Nat.Ico_zero_eq_range] end SepticProfile.FixedInterval namespace SepticProfile.EulerPolynomial open Filter Set Polynomial open scoped Topology ContDiff lemma eval_add_monomial (p : ℝ[X]) (δ x : ℝ) : (p+C δ*X^140).eval x=p.eval x+δ*x^140 := by rw [eval_add] congr 1 rw [eval_mul,eval_C,eval_pow,eval_X] lemma eval_barrier (f : PowerSeries ℝ) (δ x : ℝ) : (barrier f δ).eval x=(PowerSeries.trunc 221 f).eval x+δ*x^140 := by unfold barrier exact eval_add_monomial (PowerSeries.trunc 221 f) δ x lemma taylor_jet_littleO (f : ℝ → ℝ) (hf : ContDiffAt ℝ 220 f 0) : (fun x => f x-(PowerSeries.trunc 221 (SonicJet.jet f)).eval x) =o[𝓝 (0 : ℝ)] (fun x => x^220) := by obtain ⟨s,hs,hfs⟩ := hf.contDiffOn le_rfl (by simp) obtain ⟨r,hr,hrs⟩ := Metric.mem_nhds_iff.mp hs have hfb : ContDiffOn ℝ 220 f (Metric.ball 0 r) := hfs.mono hrs have h0 : (0 : ℝ) ∈ Metric.ball 0 r := Metric.mem_ball_self hr have ht := taylor_isLittleO (convex_ball (0 : ℝ) r) h0 hfb rw [nhdsWithin_eq_nhds.mpr (Metric.ball_mem_nhds _ hr)] at ht have heval (x : ℝ) : taylorWithinEval f 220 (Metric.ball 0 r) 0 x= (PowerSeries.trunc 221 (SonicJet.jet f)).eval x := by rw [taylor_within_apply] change (∑ k ∈ Finset.range 221, ((k.factorial : ℝ)⁻¹*(x-0)^k) • iteratedDerivWithin k f (Metric.ball 0 r) 0)= (PowerSeries.trunc 221 (SonicJet.jet f)).eval₂ (RingHom.id ℝ) x rw [PowerSeries.eval₂_trunc_eq_sum_range] apply Finset.sum_congr rfl intro k hk rw [iteratedDerivWithin_of_isOpen Metric.isOpen_ball h0,SonicJet.coeff_jet] simp only [RingHom.id_apply,sub_zero,smul_eq_mul,div_eq_mul_inv] ring simpa only [heval,sub_zero] using ht /-- The barriers enter on the correct sides of the actual analytic sonic solution. No numerical assumption about the omitted Taylor tail is needed. -/ lemma exists_barrier_entry (f : ℝ → ℝ) (hf : ContDiffAt ℝ 220 f 0) (δ : ℝ) (hδ : 0<δ) : ∀ᶠ x in 𝓝 (0 : ℝ), 0 linarith only [hbound'.1,hbound'.2,hmul,hpos] end SepticProfile.EulerPolynomial namespace SepticProfile.FixedInterval open Finset lemma tail_unscaled_bound {Q : ℤ} (hQ : 0 ha i (mem_Ico.mp hi).2) 1 50 (by norm_num) (by simpa using hall) hx hx' rw [Nat.Ico_zero_eq_range] at hAll constructor · have hh := (abs_le.mp hAll).1 norm_num at hh ⊢ exact hh have hTail := tail_unscaled_bound hQ 8 63 220 (by norm_num) b a (fun i hi => ha i (mem_Ico.mp hi).2) 1 100000 (by norm_num) (by simpa using htail) hx hx' have hMid : (∑ i ∈ Ico 1 63, a i*x^i)≤0 := by apply sum_nonpos intro i hi have hm := hmid i (mem_Ico.mp hi).1 (mem_Ico.mp hi).2 have hmi : ((b i).center : ℝ)+(b i).radius≤0 := by exact_mod_cast hm have hb := (bounds (ha i (by have := (mem_Ico.mp hi).2; omega))).2 have hq : 0<(Q : ℝ) := by exact_mod_cast hQ have hai : a i≤0 := by nlinarith exact mul_nonpos_of_nonpos_of_nonneg hai (pow_nonneg hx _) have hsum := Finset.sum_Ico_consecutive (fun i => a i*x^i) (by norm_num : 0≤1) (by norm_num : 1≤63) have hsum' := Finset.sum_Ico_consecutive (fun i => a i*x^i) (by norm_num : 0≤63) (by norm_num : 63≤220) simp only [Nat.Ico_zero_eq_range] at hsum hsum' norm_num at hsum have ht := (abs_le.mp hTail).2 norm_num at ht linarith end SepticProfile.FixedInterval namespace SepticProfile.EulerPolynomial open Polynomial lemma eval_shift_one (p : ℝ[X]) (hp : p.natDegree≤220) (h0 : p.coeff 0=1) (x : ℝ) : p.eval x=1+x*∑ i ∈ Finset.range 220, p.coeff (i+1)*x^i := by rw [eval_eq_sum_range' (show p.natDegree<221 by omega), show 221=1+220 by norm_num,Finset.sum_range_add] simp only [Finset.sum_range_one,h0,pow_zero,mul_one,Finset.mul_sum] congr 1 apply Finset.sum_congr rfl intro i hi rw [Nat.add_comm 1 i,pow_succ] ring lemma polynomial_velocity_range {Q : ℤ} (hQ : 0 p.coeff (i+1)) ha (by simpa only [zero_add] using hfirst) hmid hall htail hx hx' rw [eval_shift_one p hp h0 x] constructor · nlinarith [mul_le_mul_of_nonneg_left hh.1 hx] · nlinarith [mul_le_mul_of_nonneg_left hh.2 hx] end SepticProfile.EulerPolynomial namespace SepticProfile.RegularContinuation open Set open scoped NNReal /-- Projection to a closed scalar interval. It is used only away from the source singularities, and is proved inactive by exact comparison barriers. -/ def clamp (c d x : ℝ) : ℝ := max (min x d) c lemma clamp_mem {c d : ℝ} (hcd : c≤d) (x : ℝ) : clamp c d x ∈ Icc c d := by exact ⟨le_max_right _ _,max_le (min_le_right _ _) hcd⟩ lemma clamp_eq {c d x : ℝ} (hx : x ∈ Icc c d) : clamp c d x=x := by simp [clamp,min_eq_left hx.2,max_eq_left hx.1] lemma clamp_lipschitz (c d : ℝ) : LipschitzWith 1 (clamp c d) := (LipschitzWith.id.min_const d).max_const c /-- An ordinary (nonsingular) ODE on a compact rectangle has a clamped solution on the entire time interval. No singular-point continuation is imported here. -/ theorem exists_clamped_solution {a b c d : ℝ} (hab : a≤b) (hcd : c≤d) {f : ℝ×ℝ → ℝ} (hf : ContDiffOn ℝ 1 f (Icc a b ×ˢ Icc c d)) (u0 : ℝ) : ∃ u : ℝ → ℝ, u a=u0 ∧ ContinuousOn u (Icc a b) ∧ ∀ t ∈ Icc a b, HasDerivWithinAt u (f (t,clamp c d (u t))) (Icc a b) t := by let S := Icc a b ×ˢ Icc c d have hcomp : IsCompact S := isCompact_Icc.prod isCompact_Icc obtain ⟨K,hK⟩ := hf.exists_lipschitzOnWith (by norm_num) ((convex_Icc a b).prod (convex_Icc c d)) hcomp obtain ⟨B,hB⟩ := hcomp.exists_bound_of_continuousOn hf.continuousOn let L : ℝ≥0 := ⟨max B 0,le_max_right _ _⟩ let A : ℝ≥0 := ⟨(L:ℝ)*(b-a)+1,by have := L.coe_nonneg; nlinarith⟩ let F : ℝ → ℝ → ℝ := fun t u => f (t,clamp c d u) let t0 : Icc a b := ⟨a,le_rfl,hab⟩ have hLip (t : ℝ) (ht : t ∈ Icc a b) : LipschitzWith K (F t) := by rw [lipschitzWith_iff_norm_sub_le] intro x y have hb1 := hK.norm_sub_le (show (t,clamp c d x) ∈ S from ⟨ht,clamp_mem hcd x⟩) (show (t,clamp c d y) ∈ S from ⟨ht,clamp_mem hcd y⟩) have hb2 := (clamp_lipschitz c d).norm_sub_le x y simp only [Prod.norm_def,Prod.fst_sub,Prod.snd_sub,sub_self,norm_zero, max_eq_right (norm_nonneg _),NNReal.coe_one,one_mul] at hb1 hb2 exact hb1.trans (mul_le_mul_of_nonneg_left hb2 K.coe_nonneg) have hPic : IsPicardLindelof F t0 u0 A 0 L K := by refine ⟨fun t ht => (hLip t ht).lipschitzOnWith,?_,?_,?_⟩ · intro u _ exact hf.continuousOn.comp (continuousOn_id.prodMk continuousOn_const) (fun t ht => ⟨ht,clamp_mem hcd u⟩) · intro t ht u _ exact (hB (t,clamp c d u) ⟨ht,clamp_mem hcd u⟩).trans (le_max_left _ _) · have hA : (A : ℝ)=(L : ℝ)*(b-a)+1 := rfl change (L : ℝ)*max (b-a) (a-a) ≤ (A : ℝ)-(0 : ℝ) rw [hA,sub_self,max_eq_left (sub_nonneg.mpr hab),sub_zero] linarith obtain ⟨u,hu0,hud⟩ := hPic.exists_eq_forall_mem_Icc_hasDerivWithinAt (x := u0) (by simp) exact ⟨u,hu0,fun t ht => (hud t ht).continuousWithinAt,hud⟩ end SepticProfile.RegularContinuation namespace SepticProfile.RegularContinuation open Set /-- A compact nonsingular scalar ODE is continued between verified strict barriers. The integrator is constructed, rather than assumed. -/ theorem exists_between_strict_barriers {a b c d u0 : ℝ} (hab : a≤b) (hcd : c≤d) {f : ℝ×ℝ → ℝ} (hf : ContDiffOn ℝ 1 f (Icc a b ×ˢ Icc c d)) (lo hi dlo dhi : ℝ → ℝ) (hloder : ∀ t ∈ Icc a b, HasDerivAt lo (dlo t) t) (hhider : ∀ t ∈ Icc a b, HasDerivAt hi (dhi t) t) (hlorange : ∀ t ∈ Icc a b, lo t ∈ Icc c d) (hhirange : ∀ t ∈ Icc a b, hi t ∈ Icc c d) (hlostrict : ∀ t ∈ Icc a b, dlo t (hhider t ht).continuousAt.continuousWithinAt) (fun t ht => (hhider t ⟨ht.1,ht.2.le⟩).hasDerivWithinAt) intro t ht he rw [he,clamp_eq (hhirange t ⟨ht.1,ht.2.le⟩)] exact hhistrict t ⟨ht.1,ht.2.le⟩ have hlower : ∀ t ∈ Icc a b, lo t≤u t := by have hh := image_le_of_deriv_right_lt_deriv_boundary' huc.neg (fun t ht => (hright t ht).neg) (B := fun t => -lo t) (B' := fun t => -dlo t) (by change -u a ≤ -lo a; rw [hu]; linarith [hu0.1]) (fun t ht => (hloder t ht).neg.continuousAt.continuousWithinAt) (fun t ht => (hloder t ⟨ht.1,ht.2.le⟩).neg.hasDerivWithinAt) have hh' : ∀ t ∈ Icc a b, -u t≤ -lo t := by apply hh intro t ht he have he' : u t=lo t := neg_inj.mp he rw [he',clamp_eq (hlorange t ⟨ht.1,ht.2.le⟩)] exact neg_lt_neg (hlostrict t ⟨ht.1,ht.2.le⟩) intro t ht linarith [hh' t ht] refine ⟨u,hu,huc,fun t ht => ⟨hlower t ht,hupper t ht⟩,?_⟩ intro t ht have htu : u t ∈ Icc c d := ⟨le_trans (hlorange t ht).1 (hlower t ht),le_trans (hupper t ht) (hhirange t ht).2⟩ simpa only [clamp_eq htu] using hud t ht end SepticProfile.RegularContinuation namespace SepticProfile.EulerContinuation open Set Polynomial RegularContinuation def D (sigma t u : ℝ) : ℝ := (1-(1+1/500)*t)*((1-t)^2-sigma*(1-(1+1/500)*t)^2)*(1-u^2) def N (kappa t u : ℝ) : ℝ := -(1/500)*(1-(3/5)*u^2)*(kappa*(1-(1+1/500)*t)*(1-u^2)+ 3*((1-(1+1/500)*t)-(1-t)*u)) lemma D_pos {sigma t u : ℝ} (hs : sigma<1) (ht : t ∈ Icc 0 (5/6)) (hu : u ∈ Ico 0 1) : 0 N kappa p.1 p.2 / D sigma p.1 p.2) (Icc a b ×ˢ Icc 0 d) := by have hN : ContDiff ℝ 1 (fun p : ℝ×ℝ => N kappa p.1 p.2) := by unfold N; fun_prop have hD : ContDiff ℝ 1 (fun p : ℝ×ℝ => D sigma p.1 p.2) := by unfold D; fun_prop exact hN.contDiffOn.div hD.contDiffOn (fun p hp => ne_of_gt (D_pos hs ⟨ha.trans hp.1.1,hp.1.2.trans hb⟩ ⟨hp.2.1,hp.2.2.trans_lt hd⟩)) lemma residual_eq (sigma kappa : ℝ) (f : ℝ → ℝ) (x : ℝ) : EulerFormal.profileResidual sigma kappa (3/5) (1/500) f x= D sigma x (f x)*deriv f x-N kappa x (f x) := by unfold EulerFormal.profileResidual D N ring /-- Genuine regular continuation of the sonic Euler equation from a>0 to 5/6, provided the two polynomial residual certificates are supplied. -/ theorem exists_between_polynomials {sigma kappa a u0 : ℝ} (hs : sigma<1) (ha : 0 p.eval x) t<0) (hresq : ∀ t ∈ Ioc 0 (5/6), 0 q.eval x) t) (hu0 : u0 ∈ Icc (p.eval a) (q.eval a)) : ∃ u : ℝ → ℝ, u a=u0 ∧ ContinuousOn u (Icc a (5/6)) ∧ (∀ t ∈ Icc a (5/6), p.eval t≤u t ∧ u t≤q.eval t) ∧ ∀ t ∈ Icc a (5/6), HasDerivWithinAt u (N kappa t (u t)/D sigma t (u t)) (Icc a (5/6)) t := by have hd : (0:ℝ)≤1-a/1000 := by linarith have hd' : 1-a/1000<1 := by linarith have ht0 {t : ℝ} (ht : t ∈ Icc a (5/6)) : t ∈ Icc 0 (5/6) := ⟨ha.le.trans ht.1,ht.2⟩ have htp {t : ℝ} (ht : t ∈ Icc a (5/6)) : t ∈ Ioc 0 (5/6) := ⟨ha.trans_le ht.1,ht.2⟩ have hpr (t : ℝ) (ht : t ∈ Icc a (5/6)) : p.eval t ∈ Icc 0 (1-a/1000) := by have hh := hrangep t (ht0 ht) constructor <;> linarith [ht.1,ht.2] have hqr (t : ℝ) (ht : t ∈ Icc a (5/6)) : q.eval t ∈ Icc 0 (1-a/1000) := by have hh := hrangeq t (ht0 ht) constructor <;> linarith [ht.1,ht.2] apply exists_between_strict_barriers ha' hd (field_contDiffOn hs ha.le (le_refl (5/6)) hd') (fun t => p.eval t) (fun t => q.eval t) (fun t => deriv (fun x => p.eval x) t) (fun t => deriv (fun x => q.eval x) t) (fun t _ => p.differentiableAt.hasDerivAt) (fun t _ => q.differentiableAt.hasDerivAt) hpr hqr ?_ ?_ hu0 · intro t ht apply (lt_div_iff₀ (D_pos hs (ht0 ht) ⟨(hpr t ht).1,(hpr t ht).2.trans_lt hd'⟩)).mpr have hh := hresp t (htp ht) rw [residual_eq] at hh nlinarith only [hh] · intro t ht apply (div_lt_iff₀ (D_pos hs (ht0 ht) ⟨(hqr t ht).1,(hqr t ht).2.trans_lt hd'⟩)).mpr have hh := hresq t (htp ht) rw [residual_eq] at hh nlinarith only [hh] end SepticProfile.EulerContinuation end end end OAI