import Mathlib import OAI.Geometry.BallPacking.Layers.SmoothFormTransport namespace OAI noncomputable section namespace PackingSufficiencySupport.Hamiltonian section open scoped ContDiff Topology BigOperators open Set Function variable {ι : Type*} [Fintype ι] [DecidableEq ι] def splitCapacityBall (r : ℝ) : Set (Plane × PlanePhase ι) := {z | radialArea z.1 + ∑ i, planeMoments z.2 i ≤ r} private theorem inverse_layer_product_pullback {F : (ι → ℝ) → ℝ} {g : Plane × PlanePhase ι → Plane × PlanePhase ι} {f₀ : Plane × PlanePhase ι → ℝ} (hF : ContDiff ℝ ∞ F) (hf : ContDiff ℝ ∞ f₀) {U W : Set (Plane × PlanePhase ι)} (hU : IsOpen U) (hW : IsOpen W) (hg : ContDiffOn ℝ ∞ g W) (hr : ∀ y ∈ W, g y ∈ U ∧ layerChart F f₀ (g y) = y) {q : Plane ≃ₜ Plane} (hqs : ContDiff ℝ ∞ q) (hqform : ∀ x v w, planarArea (fderiv ℝ q x v) (fderiv ℝ q x w) = planarArea v w) {z : Plane × PlanePhase ι} (hz : planarProduct q z ∈ W) (v w : Plane × PlanePhase ι) : layerForm F f₀ ((g ∘ planarProduct q) z) (fderiv ℝ (g ∘ planarProduct q) z v) (fderiv ℝ (g ∘ planarProduct q) z w) = productForm phaseArea v w := by exact inverse_chart_composition_pullback (E := Plane × PlanePhase ι) (f := layerChart F f₀) (g := g) (q := (planarProduct q : Plane × PlanePhase ι → Plane × PlanePhase ι)) (U := U) (W := W) hU hW (layerChart_smooth hF hf).contDiffOn hg hr (layerForm F f₀) (productForm phaseArea) (fun x _ u t => layerChart_pullback hF hf x u t) (planarProduct_smooth (E := PlanePhase ι) (q := q) hqs) (fun x u t => planarProduct_pullback (E := PlanePhase ι) (q := q) hqs hqform (phaseArea (ι := ι)) x u t) hz v w theorem exists_ball_in_layer {F : (ι → ℝ) → ℝ} {f : Plane × PlanePhase ι → ℝ} (hF : ContDiff ℝ ∞ F) (hf : ContDiff ℝ ∞ f) {V : Set (ι → ℝ)} (hV : IsOpen (planeMoments ⁻¹' V)) {a b : ℝ} (hab : a < b) (H : (ι → ℝ) → ℝ) (hderiv : ∀ T ∈ Ioo (0:ℝ) 1, ∀ v, planeMoments v ∈ V → ∀ s ∈ Ioo a b, 0 < deriv (fun r => f ((r,T),v)) s) (hlo : ∀ T ∈ Ioo (0:ℝ) 1, ∀ v, planeMoments v ∈ V → f ((a,T),v) = 0) (hhi : ∀ T ∈ Ioo (0:ℝ) 1, ∀ v, planeMoments v ∈ V → H (planeMoments v) < f ((b,T),v)) {r r' : ℝ} (hr' : 0 ≤ r') (hrr : r' < r) (hsimplex : ∀ p : ι → ℝ, (∀ i, 0 ≤ p i) → (∑ i, p i) ≤ r' → p ∈ V) (hheight : ∀ p : ι → ℝ, (∀ i, 0 ≤ p i) → p ∈ V → r-∑ i, p i ≤ H p) : ∃ (N : Set (Plane × PlanePhase ι)) (φ : Plane × PlanePhase ι → Plane × PlanePhase ι), IsOpen N ∧ splitCapacityBall r' ⊆ N ∧ ContDiffOn ℝ ∞ φ N ∧ Topology.IsEmbedding (fun z : N => φ z) ∧ (∀ z ∈ N, φ z ∈ (Ioo a b ×ˢ Ioo (0:ℝ) 1) ×ˢ (planeMoments ⁻¹' V)) ∧ (∀ z ∈ N, ∀ v w, layerForm F f (φ z) (fderiv ℝ φ z v) (fderiv ℝ φ z w) = productForm phaseArea v w) := by let U := (Ioo a b ×ˢ Ioo (0:ℝ) 1) ×ˢ (planeMoments ⁻¹' V) have hU : IsOpen U := (isOpen_Ioo.prod isOpen_Ioo).prod hV obtain ⟨W,g,hW,hg,himage,ha,hl,hr⟩ := layerChart_inverse hF hf hV hab H hderiv hlo hhi let δ := (r-r')/2 have hδ : 0 < δ := by dsimp [δ]; linarith have hδr : r'+δ < r := by dsimp [δ]; linarith obtain ⟨q,hqs,_,hqform,hqbound⟩ := exists_thin_planar_embedding hr' hδ let Q : (Plane × PlanePhase ι) ≃ₜ (Plane × PlanePhase ι) := planarProduct q have hQ : ContDiff ℝ ∞ Q := planarProduct_smooth hqs let N := Q ⁻¹' W let φ := g ∘ Q have hN : IsOpen N := hW.preimage Q.continuous have hball : splitCapacityBall r' ⊆ N := by intro z hz have hsum : 0 ≤ ∑ i, planeMoments z.2 i := Finset.sum_nonneg (fun i _ => planeMoments_nonneg z.2 i) have hz' : radialArea z.1 + ∑ i, planeMoments z.2 i ≤ r' := hz have hrad : radialArea z.1 ≤ r' := by linarith have hps : (∑ i, planeMoments z.2 i) ≤ r' := by linarith [radialArea_nonneg z.1] have hpV := hsimplex (planeMoments z.2) (planeMoments_nonneg z.2) hps have hqb := hqbound z.1 hrad apply ha (Q z) hqb.2.2 hpV hqb.1 change (q z.1).1 < H (planeMoments z.2) have hh := hheight (planeMoments z.2) (planeMoments_nonneg z.2) hpV linarith [hqb.2.1] have hsφ : ContDiffOn ℝ ∞ φ N := hg.comp hQ.contDiffOn (fun _ hz => hz) have hge : Topology.IsEmbedding (fun y : W => g y) := inverse_chart_isEmbedding (layerChart_smooth hF hf).continuous.continuousOn hg.continuousOn himage.symm hl hr let j : N → W := fun z => ⟨Q z,z.property⟩ have hje : Topology.IsEmbedding j := (Q.isEmbedding.comp Topology.IsEmbedding.subtypeVal).codRestrict W (fun z => z.property) refine ⟨N,φ,hN,hball,hsφ,hge.comp hje,?_,?_⟩ · intro z hz exact (hr (Q z) hz).1 · intro z hz v w exact inverse_layer_product_pullback hF hf hU hW hg hr hqs hqform hz v w end section open scoped ContDiff Topology BigOperators open Set Function def complexSplitPhaseLinear (m : ℕ) : Ambient (m+1) ≃ₗ[ℝ] Plane × PlanePhase (Fin m) where toFun z := (((z 0).re,(z 0).im),fun j => ((z j.succ).re,(z j.succ).im)) invFun p := Fin.cons (⟨p.1.1,p.1.2⟩ : ℂ) (fun j => (⟨(p.2 j).1,(p.2 j).2⟩ : ℂ)) left_inv z := by funext i; exact Fin.cases rfl (fun j => rfl) i right_inv p := rfl map_add' _ _ := rfl map_smul' a z := by ext <;> simp [Pi.smul_apply] def complexSplitPhase (m : ℕ) : Ambient (m+1) ≃L[ℝ] Plane × PlanePhase (Fin m) := (complexSplitPhaseLinear m).toContinuousLinearEquiv @[simp] theorem complexSplitPhase_apply (m : ℕ) (z : Ambient (m+1)) : complexSplitPhase m z=(((z 0).re,(z 0).im),fun j => ((z j.succ).re,(z j.succ).im)) := rfl theorem complexSplitPhase_capacity (m : ℕ) (z : Ambient (m+1)) : radialArea (complexSplitPhase m z).1 + ∑ j,planeMoments (complexSplitPhase m z).2 j = capacity z := by simp only [complexSplitPhase_apply,radialArea,radiusSq,planeMoments,capacity,Fin.sum_univ_succ, Complex.normSq_apply,Finset.mul_sum,mul_add,pow_two] theorem complexSplitPhase_form (m : ℕ) (v w : Ambient (m+1)) : productForm phaseArea (complexSplitPhase m v) (complexSplitPhase m w)=standardForm v w := by simp only [productForm_apply,phaseArea_apply,complexSplitPhase_apply,standardForm,Fin.sum_univ_succ] rfl @[simp] theorem complexSplitPhase_preimage_ball (m : ℕ) (r : ℝ) : complexSplitPhase m ⁻¹' splitCapacityBall r=closedBall (m+1) r := by ext z change radialArea (complexSplitPhase m z).1+∑ j,planeMoments (complexSplitPhase m z).2 j ≤ r ↔ capacity z ≤ r rw [complexSplitPhase_capacity] theorem complex_ball_of_split_embedding {m : ℕ} {r : ℝ} {Ω : Plane × PlanePhase (Fin m) → (Plane × PlanePhase (Fin m)) →L[ℝ] (Plane × PlanePhase (Fin m)) →L[ℝ] ℝ} {N : Set (Plane × PlanePhase (Fin m))} {φ : Plane × PlanePhase (Fin m) → Plane × PlanePhase (Fin m)} (hN : IsOpen N) (hball : splitCapacityBall r ⊆ N) (hs : ContDiffOn ℝ ∞ φ N) (he : Topology.IsEmbedding (fun z : N => φ z)) (hform : ∀ z ∈ N,∀ v w,Ω (φ z) (fderiv ℝ φ z v) (fderiv ℝ φ z w)=productForm phaseArea v w) : ∃ U : Set (Ambient (m+1)),IsOpen U ∧ closedBall (m+1) r ⊆ U ∧ ContDiffOn ℝ ∞ (φ ∘ complexSplitPhase m) U ∧ Topology.IsEmbedding (fun z : U => φ (complexSplitPhase m z)) ∧ (∀ z ∈ U,φ (complexSplitPhase m z)∈φ '' N) ∧ ∀ z ∈ U,∀ v w,Ω (φ (complexSplitPhase m z)) (fderiv ℝ (φ ∘ complexSplitPhase m) z v) (fderiv ℝ (φ ∘ complexSplitPhase m) z w)=standardForm v w := by let e := complexSplitPhase m let U := e ⁻¹' N have hU : IsOpen U := hN.preimage e.continuous have hm : MapsTo e U N := fun _ hz => hz let j : U → N := fun z => ⟨e z,z.property⟩ have hj : Topology.IsEmbedding j := (e.toHomeomorph.isEmbedding.comp Topology.IsEmbedding.subtypeVal).codRestrict N (fun z => z.property) refine ⟨U,hU,?_,hs.comp e.contDiff.contDiffOn hm,he.comp hj,?_,?_⟩ · intro z hz apply hball have hz' : z∈e ⁻¹' splitCapacityBall r := by simpa only [e,complexSplitPhase_preimage_ball] using hz exact hz' · intro z hz exact mem_image_of_mem φ hz · intro z hz v w have hd := fderiv_comp z ((hs.contDiffAt (hN.mem_nhds hz)).differentiableAt (by simp)) e.differentiableAt rw [hd,e.fderiv] change Ω (φ (e z)) (fderiv ℝ φ (e z) (e v)) (fderiv ℝ φ (e z) (e w))=standardForm v w rw [hform (e z) hz,complexSplitPhase_form] end section open scoped ContDiff Topology open Set Function variable {E F Q : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup Q] [NormedSpace ℝ Q] [CompleteSpace Q] theorem parametric_inverse_on_local (f : Q × E → E) (D : Set (Q × E)) (hD : IsOpen D) (hf : ContDiffOn ℝ ∞ f D) (hinj : ∀ t, InjOn (fun x => f (t,x)) {x | (t,x) ∈ D}) (hi : ∀ p ∈ D, (fderiv ℝ (fun x => f (p.1,x)) p.2).IsInvertible) : ∃ (U : Set (Q × E)) (g : Q × E → E), IsOpen U ∧ ContDiffOn ℝ ∞ g U ∧ U = (fun p => (p.1,f p)) '' D ∧ (∀ p ∈ D, (p.1,f p) ∈ U ∧ g (p.1,f p) = p.2) ∧ (∀ p ∈ U, (p.1,g p) ∈ D ∧ f (p.1,g p) = p.2) := by let F : Q × E → Q × E := fun p => (p.1,f p) have hF : ContDiffOn ℝ ∞ F D := contDiffOn_fst.prodMk hf have hFi : InjOn F D := by rintro ⟨t,x⟩ hp ⟨s,y⟩ hq he have ht : t = s := congrArg (fun z : Q × E => z.1) he cases ht have hx : f (t,x) = f (t,y) := congrArg (fun z : Q × E => z.2) he exact Prod.ext rfl (hinj t hp hq hx) have hFd (p : Q × E) (hp : p∈D) : fderiv ℝ F p = (ContinuousLinearMap.fst ℝ Q E).prod (fderiv ℝ f p) := by exact (hasFDerivAt_fst.prodMk ((hf.contDiffAt (hD.mem_nhds hp)).differentiableAt (by simp)).hasFDerivAt).fderiv have hFiv (p : Q × E) (hp : p ∈ D) : (fderiv ℝ F p).IsInvertible := by rw [hFd p hp] apply clockGraph_linear_invertible have hd : fderiv ℝ (fun x => f (p.1,x)) p.2 = (fderiv ℝ f p).comp (ContinuousLinearMap.inr ℝ Q E) := by exact (((hf.contDiffAt (hD.mem_nhds hp)).differentiableAt (by simp)).hasFDerivAt.comp p.2 ((hasFDerivAt_const p.1 p.2).prodMk (hasFDerivAt_id p.2))).fderiv rw [← hd] exact hi p hp obtain ⟨hU,hg,hleft,hright⟩ := smooth_inverse_on hD hF hFi hFiv let g : Q × E → E := fun p => (invFunOn F D p).2 refine ⟨F '' D,g,hU,hg.snd,rfl,?_,?_⟩ · intro p hp refine ⟨⟨p,hp,rfl⟩,?_⟩ exact congrArg Prod.snd (hleft p hp) · intro p hp have he := hright p hp change ((invFunOn F D p).1,f (invFunOn F D p)) = p at he have hfirst := congrArg (fun z : Q × E => z.1) he have hmem := invFunOn_mem hp have hid : (p.1,g p) = invFunOn F D p := Prod.ext hfirst.symm rfl rw [hid] exact ⟨hmem,congrArg (fun z : Q × E => z.2) he⟩ end section open scoped ContDiff Topology open Set Function variable {Q : Type*} [NormedAddCommGroup Q] [NormedSpace ℝ Q] [CompleteSpace Q] theorem monotone_strip_inverse_local {f : Q × ℝ → ℝ} {U : Set Q} (hU : IsOpen U) {a b : ℝ} (hab : a < b) (hf : ContDiffOn ℝ ∞ f (U ×ˢ Icc a b)) {h : Q → ℝ} (hderiv : ∀ q ∈ U, ∀ s ∈ Ioo a b, 0 < deriv (fun r => f (q,r)) s) (hlo : ∀ q ∈ U, f (q,a) = 0) (hhi : ∀ q ∈ U, h q < f (q,b)) : ∃ (W : Set (Q × ℝ)) (g : Q × ℝ → ℝ), IsOpen W ∧ ContDiffOn ℝ ∞ g W ∧ W = (fun p : Q × ℝ => (p.1,f p)) '' (U ×ˢ Ioo a b) ∧ (∀ q ∈ U, ∀ x, 0 < x → x < h q → (q,x) ∈ W) ∧ (∀ q ∈ U, ∀ s ∈ Ioo a b, g (q,f (q,s)) = s) ∧ (∀ p ∈ W, p.1 ∈ U ∧ g p ∈ Ioo a b ∧ f (p.1,g p) = p.2) := by have hc (q : Q) (hq : q∈U) : ContinuousOn (fun s => f (q,s)) (Icc a b) := hf.continuousOn.comp (continuous_const.prodMk continuous_id).continuousOn (fun _ hs => ⟨hq,hs⟩) have hm (q : Q) (hq : q ∈ U) : StrictMonoOn (fun s => f (q,s)) (Icc a b) := by apply strictMonoOn_of_deriv_pos (convex_Icc a b) (hc q hq) simpa only [interior_Icc] using hderiv q hq have hinj (q : Q) : InjOn (fun s => f (q,s)) {s | (q,s) ∈ U ×ˢ Ioo a b} := by intro s hs t ht he exact (hm q hs.1).injOn ⟨hs.2.1.le,hs.2.2.le⟩ ⟨ht.2.1.le,ht.2.2.le⟩ he have hiv (p : Q × ℝ) (hp : p ∈ U ×ˢ Ioo a b) : (fderiv ℝ (fun s => f (p.1,s)) p.2).IsInvertible := by let c := deriv (fun s => f (p.1,s)) p.2 have hc0 : c ≠ 0 := ne_of_gt (hderiv p.1 hp.1 p.2 hp.2) let e : ℝ ≃L[ℝ] ℝ := (LinearEquiv.smulOfNeZero ℝ ℝ c hc0).toContinuousLinearEquiv refine ⟨e,?_⟩ ext simp [e,c] obtain ⟨W,g,hWo,hgs,hWe,hleft,hright⟩ := parametric_inverse_on_local f (U ×ˢ Ioo a b) (hU.prod isOpen_Ioo) (hf.mono (prod_mono_right Ioo_subset_Icc_self)) hinj hiv refine ⟨W,g,hWo,hgs,hWe,?_,?_,?_⟩ · intro q hq x hx0 hxh have hh : x ∈ Ioo (f (q,a)) (f (q,b)) := ⟨by rwa [hlo q hq],hxh.trans (hhi q hq)⟩ obtain ⟨s,hs,hse⟩ := intermediate_value_Ioo hab.le (hc q hq) hh rw [hWe] exact ⟨(q,s),⟨hq,hs⟩,Prod.ext rfl hse⟩ · intro q hq s hs exact (hleft (q,s) ⟨hq,hs⟩).2 · intro p hp exact ⟨(hright p hp).1.1,(hright p hp).1.2,(hright p hp).2⟩ end open scoped ContDiff Topology open Set Function variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] theorem stripChart_inverse_local {f : Plane × E → ℝ} {V : Set E} (hV : IsOpen V) {a b : ℝ} (hab : a < b) (hf : ContDiffOn ℝ ∞ f ((Icc a b ×ˢ Ioo (0:ℝ) 1) ×ˢ V)) (H : E → ℝ) (hderiv : ∀ T ∈ Ioo (0:ℝ) 1, ∀ v ∈ V, ∀ s ∈ Ioo a b, 0 < deriv (fun r => f ((r,T),v)) s) (hlo : ∀ T ∈ Ioo (0:ℝ) 1, ∀ v ∈ V, f ((a,T),v) = 0) (hhi : ∀ T ∈ Ioo (0:ℝ) 1, ∀ v ∈ V, H v < f ((b,T),v)) : ∃ (W : Set (Plane × E)) (g : Plane × E → Plane × E), IsOpen W ∧ ContDiffOn ℝ ∞ g W ∧ W = stripChart f '' ((Ioo a b ×ˢ Ioo (0:ℝ) 1) ×ˢ V) ∧ (∀ p, p.1.2 ∈ Ioo (0:ℝ) 1 → p.2 ∈ V → 0 < p.1.1 → p.1.1 < H p.2 → p ∈ W) ∧ (∀ p ∈ (Ioo a b ×ˢ Ioo (0:ℝ) 1) ×ˢ V, g (stripChart f p) = p) ∧ (∀ p ∈ W, g p ∈ (Ioo a b ×ˢ Ioo (0:ℝ) 1) ×ˢ V ∧ stripChart f (g p) = p) := by let e : (Plane × E) ≃L[ℝ] ((ℝ × E) × ℝ) := layerOrder let f' : (ℝ × E) × ℝ → ℝ := f ∘ e.symm have hf' : ContDiffOn ℝ ∞ f' ((Ioo (0:ℝ) 1 ×ˢ V) ×ˢ Icc a b) := hf.comp e.symm.contDiff.contDiffOn (fun _ hp => ⟨⟨hp.2,hp.1.1⟩,hp.1.2⟩) obtain ⟨W,g,hWo,hgs,hWe,havailable,hleft,hright⟩ := monotone_strip_inverse_local (isOpen_Ioo.prod hV) hab hf' (h := fun q : ℝ × E => H q.2) (fun q hq s hs => hderiv q.1 hq.1 q.2 hq.2 s hs) (fun q hq => hlo q.1 hq.1 q.2 hq.2) (fun q hq => hhi q.1 hq.1 q.2 hq.2) let G : Plane × E → Plane × E := fun p => ((g (e p),p.1.2),p.2) refine ⟨e ⁻¹' W,G,hWo.preimage e.continuous,?_,?_,?_,?_,?_⟩ · have hg : ContDiffOn ℝ ∞ (fun p => g (e p)) (e ⁻¹' W) := hgs.comp e.contDiff.contDiffOn (fun _ hp => hp) exact (hg.prodMk (contDiff_snd.comp contDiff_fst).contDiffOn).prodMk contDiff_snd.contDiffOn · ext p constructor · intro hp obtain ⟨q,hq,hqe⟩ := by simpa only [hWe] using hp refine ⟨e.symm q,⟨⟨hq.2,hq.1.1⟩,hq.1.2⟩,?_⟩ apply e.injective change (q.1,f' q) = e p exact hqe · rintro ⟨q,hq,rfl⟩ change e (stripChart f q) ∈ W rw [hWe] exact ⟨e q,⟨⟨hq.1.2,hq.2⟩,hq.1.1⟩,rfl⟩ · intro p hpT hpV hp0 hpH exact havailable (p.1.2,p.2) ⟨hpT,hpV⟩ p.1.1 hp0 hpH · intro p hp have hh := hleft (p.1.2,p.2) ⟨hp.1.2,hp.2⟩ p.1.1 hp.1.1 change ((g ((p.1.2,p.2),f p),p.1.2),p.2) = p change g ((p.1.2,p.2),f p) = p.1.1 at hh rw [hh] · intro p hp have hh := hright (e p) hp refine ⟨⟨⟨hh.2.1,hh.1.1⟩,hh.1.2⟩,?_⟩ change ((f ((g (e p),p.1.2),p.2),p.1.2),p.2) = p have he : f ((g (e p),p.1.2),p.2) = p.1.1 := hh.2.2 rw [he] end PackingSufficiencySupport.Hamiltonian end end OAI