import Mathlib import OAI.Geometry.BallPacking.Annuli.CylinderAnnularNormalization namespace OAI noncomputable section namespace PackingSufficiencySupport.Hamiltonian section open scoped ContDiff Topology open Set Function variable {ι : Type*} [Fintype ι] [DecidableEq ι] def layerHamiltonian (F H : (ι → ℝ) → ℝ) (ρ : ℝ → ℝ) (p : ℝ × PlanePhase ι) : ℝ := F (planeMoments p.2)+ρ p.1*H (planeMoments p.2) omit [DecidableEq ι] in @[fun_prop] theorem layerHamiltonian_smooth {F H : (ι → ℝ) → ℝ} {ρ : ℝ → ℝ} (hF : ContDiff ℝ ∞ F) (hH : ContDiff ℝ ∞ H) (hρ : ContDiff ℝ ∞ ρ) : ContDiff ℝ ∞ (layerHamiltonian F H ρ) := (hF.comp (planeMoments_smooth.comp contDiff_snd)).add ((hρ.comp contDiff_fst).mul (hH.comp (planeMoments_smooth.comp contDiff_snd))) omit [DecidableEq ι] in private theorem scalar_area_slice {L : Plane → ℝ} {q : Plane} {c : ℝ} (hL : DifferentiableAt ℝ L q) (hc : HasDerivAt (fun s => L (s,q.2)) c q.1) : fderiv ℝ L q (1,0)=c := by have hline := (hasDerivAt_id q.1).prodMk (hasDerivAt_const q.1 q.2) exact (hL.hasFDerivAt.comp_hasDerivAt q.1 hline).unique hc omit [DecidableEq ι] in theorem areaLayerForm_apply {I : Set ℝ} (hI : IsOpen I) (hconv : Convex ℝ I) {a : ℝ} (ha : a∈I) {ℓ : Plane → ℝ} (hℓ : ContDiffOn ℝ ∞ ℓ (I ×ˢ Ioo (0:ℝ) 1)) {F H : (ι → ℝ) → ℝ} {ρ : ℝ → ℝ} (hF : ContDiff ℝ ∞ F) (hH : ContDiff ℝ ∞ H) (hρ : ContDiff ℝ ∞ ρ) {p : Plane × PlanePhase ι} (hp : p.1∈I ×ˢ Ioo (0:ℝ) 1) (v w : Plane × PlanePhase ι) : layerForm F (areaLayerCoordinate ℓ a ρ H) p v w=phaseArea v.2 w.2+ ℓ p.1*(v.1.1*w.1.2-v.1.2*w.1.1)+ fderiv ℝ (layerHamiltonian F H ρ) (p.1.1,p.2) (v.1.1,v.2)*w.1.2- v.1.2*fderiv ℝ (layerHamiltonian F H ρ) (p.1.1,p.2) (w.1.1,w.2) := by let L := layerAreaIntegral ℓ a let K : ℝ × PlanePhase ι → ℝ := fun q => ρ q.1*H (planeMoments q.2) have hL : DifferentiableAt ℝ L p.1 := ((layerAreaIntegral_smooth hI hconv ha hℓ).contDiffAt ((hI.prod isOpen_Ioo).mem_nhds hp)).differentiableAt (by simp) have hK : Differentiable ℝ K := ((hρ.comp contDiff_fst).mul (hH.comp (planeMoments_smooth.comp contDiff_snd))).differentiable (by simp) have hFs : Differentiable ℝ (toricFunction F) := (hF.comp planeMoments_smooth).differentiable (by simp) have hpart : fderiv ℝ L p.1 (1,0)=ℓ p.1 := scalar_area_slice hL (layerAreaIntegral_hasDerivAt hI hconv ha hp.1 hp.2 hℓ) have hfd (u : Plane × PlanePhase ι) : fderiv ℝ (areaLayerCoordinate ℓ a ρ H) p u= fderiv ℝ L p.1 u.1+fderiv ℝ K (p.1.1,p.2) (u.1.1,u.2) := by change fderiv ℝ (baseCoefficient L+firstBaseCoefficient K) p u=_ have hLD : DifferentiableAt ℝ (baseCoefficient L) p := hL.comp p differentiableAt_fst have hKD : DifferentiableAt ℝ (firstBaseCoefficient K) p := (hK _).comp p ((differentiableAt_fst.fst).prodMk differentiableAt_snd) rw [fderiv_add hLD hKD] simp only [add_apply,baseCoefficient_fderiv_at hL,firstBaseCoefficient_fderiv_at (hK _)] have hgd (u : ℝ × PlanePhase ι) : fderiv ℝ (layerHamiltonian F H ρ) (p.1.1,p.2) u= fderiv ℝ (toricFunction F) p.2 u.2+fderiv ℝ K (p.1.1,p.2) u := by change fderiv ℝ ((toricFunction F ∘ Prod.snd)+K) (p.1.1,p.2) u=_ rw [fderiv_add ((hFs _).comp _ differentiableAt_snd) (hK _), fderiv_comp _ (hFs _) differentiableAt_snd] simp only [add_apply,ContinuousLinearMap.comp_apply,fderiv_snd] rfl rw [layerForm_apply,hfd,hfd,hgd,hgd,plane_covector_decomp (fderiv ℝ L p.1) v.1, plane_covector_decomp (fderiv ℝ L p.1) w.1,hpart] ring end section open scoped ContDiff Topology BigOperators open Set Function def areaHamiltonianForm {m : ℕ} (ℓ : Plane → ℝ) (G : ℝ × PlanePhase (Fin m) → ℝ) (p : Plane × PlanePhase (Fin m)) : (Plane × PlanePhase (Fin m)) →L[ℝ] (Plane × PlanePhase (Fin m)) →L[ℝ] ℝ := let ds := (ContinuousLinearMap.fst ℝ ℝ ℝ).comp (ContinuousLinearMap.fst ℝ Plane (PlanePhase (Fin m))) let dt := (ContinuousLinearMap.snd ℝ ℝ ℝ).comp (ContinuousLinearMap.fst ℝ Plane (PlanePhase (Fin m))) let c := ds.prod (ContinuousLinearMap.snd ℝ Plane (PlanePhase (Fin m))) let β := (fderiv ℝ G (p.1.1,p.2)).comp c phaseArea.bilinearComp (ContinuousLinearMap.snd ℝ Plane (PlanePhase (Fin m))) (ContinuousLinearMap.snd ℝ Plane (PlanePhase (Fin m)))+ ℓ p.1 • (ds.smulRight dt-dt.smulRight ds)+β.smulRight dt-dt.smulRight β @[simp] theorem areaHamiltonianForm_apply {m : ℕ} (ℓ : Plane → ℝ) (G : ℝ × PlanePhase (Fin m) → ℝ) (p v w : Plane × PlanePhase (Fin m)) : areaHamiltonianForm ℓ G p v w=phaseArea v.2 w.2+ℓ p.1*(v.1.1*w.1.2-v.1.2*w.1.1)+ fderiv ℝ G (p.1.1,p.2) (v.1.1,v.2)*w.1.2-v.1.2*fderiv ℝ G (p.1.1,p.2) (w.1.1,w.2) := by simp [areaHamiltonianForm,ContinuousLinearMap.smulRight_apply,smul_eq_mul] theorem areaHamiltonianForm_layer {m : ℕ} {I : Set ℝ} (hI : IsOpen I) (hc : Convex ℝ I) {a b : ℝ} (ha : a∈I) (hb : b∈I) (hab : a Disjoint (Ioo (a i) (b i)) (Ioo (a j) (b j)))) (hr' : ∀ i,0≤r' i) (hrr : ∀ i,r' i φ i z)) ∧ (∀ i,∀ z∈U i,φ i z∈(Ioo (a i) (b i) ×ˢ Ioo (0:ℝ) 1) ×ˢ (planeMoments ⁻¹' V)) ∧ (∀ i,∀ z∈U i,∀ v w,areaHamiltonianForm ℓ G (φ i z) (fderiv ℝ (φ i) z v) (fderiv ℝ (φ i) z w)=standardForm v w) ∧ Pairwise (fun i j => Disjoint (φ i '' closedBall (m+1) (r' i)) (φ j '' closedBall (m+1) (r' j))) := by have hd i := areaLayerCoordinate_data hI hc (ha i) (hb i) (hab i) hℓ hpos (hρ i) (hρpos i) (hρa i) (hρb i) (hH i) (hHp i) exact finite_layer_packing_local F H (fun i => areaLayerCoordinate ℓ (a i) (ρ i) (H i)) hF hV a b r r' hab (fun i => (hd i).1) hdis (fun i => (hd i).2.1) (fun i => (hd i).2.2.1) (fun i => (hd i).2.2.2) hr' hrr hsimplex hheight (areaHamiltonianForm ℓ G) (fun i _ hp => areaHamiltonianForm_layer hI hc (ha i) (hb i) (hab i) hℓ (hF i) (hH i) (hρ i) hV (he i) hp) end section open scoped ContDiff Topology open Set Function theorem normalizedAreaDensity_smooth {I J : Set ℝ} (hI : IsOpen I) (hJ : IsOpen J) {A B : Plane → ℝ} (hA : ContDiffOn ℝ ∞ A (I ×ˢ J)) (hB : ContDiffOn ℝ ∞ B (I ×ˢ J)) {g : ℝ → ℝ} (hg : ContDiffOn ℝ ∞ g (Ioo (0:ℝ) 1)) (hmap : MapsTo g (Ioo (0:ℝ) 1) J) : ContDiffOn ℝ ∞ (normalizedAreaDensity A B g) (I ×ˢ Ioo (0:ℝ) 1) := by have hq : ContDiffOn ℝ ∞ (fun q : Plane => (q.1,g q.2)) (I ×ˢ Ioo (0:ℝ) 1) := contDiffOn_fst.prodMk (hg.comp contDiffOn_snd (fun _ h => h.2)) have hqm : MapsTo (fun q : Plane => (q.1,g q.2)) (I ×ˢ Ioo (0:ℝ) 1) (I ×ˢ J) := fun _ h => ⟨h.1,hmap h.2⟩ exact ((((hB.fderiv_of_isOpen (hI.prod hJ) (by simp)).comp hq hqm).clm_apply contDiffOn_const).sub (((hA.fderiv_of_isOpen (hI.prod hJ) (by simp)).comp hq hqm).clm_apply contDiffOn_const)).mul ((hg.deriv_of_isOpen isOpen_Ioo (by simp)).comp contDiffOn_snd (fun _ h => h.2)) theorem normalizedAreaDensity_pos {I J : Set ℝ} {A B : Plane → ℝ} {g : ℝ → ℝ} (hmap : MapsTo g (Ioo (0:ℝ) 1) J) (hg : ∀ T∈Ioo (0:ℝ) 1,0 hm _ h,hl,hr,fun T hT => (hd T hT).2, normalizedAreaDensity_smooth hI isOpen_Ioo hA hB hg (fun _ h => hm _ h),?_⟩ intro q hq exact normalizedAreaDensity_pos (fun _ h => hm _ h) (fun T hT => (hd T hT).1) hbase hq end section open scoped ContDiff Topology open Set Function variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] theorem angularReparam_smoothOn {g : ℝ → ℝ} {J : Set ℝ} (hg : ContDiffOn ℝ ∞ g J) {U : Set (Plane × E)} (hU : ∀ p∈U,p.1.2∈J) : ContDiffOn ℝ ∞ (angularReparam g) U := ((contDiff_fst.comp contDiff_fst).contDiffOn.prodMk (hg.comp (contDiff_snd.comp contDiff_fst).contDiffOn hU)).prodMk contDiff_snd.contDiffOn theorem angularReparam_isEmbedding {g C : ℝ → ℝ} {J : Set ℝ} (hg : ContDiffOn ℝ ∞ g J) (hC : Continuous C) (hinv : ∀ T∈J,C (g T)=T) {U : Set (Plane × E)} (hU : ∀ p∈U,p.1.2∈J) : Topology.IsEmbedding (fun p : U => angularReparam g p.val) := by have hs : Continuous (fun p : U => angularReparam g p.val) := (angularReparam_smoothOn hg hU).continuousOn.domRestrict have hCs : Continuous (angularReparam (E := E) C) := (continuous_fst.fst.prodMk (hC.comp continuous_fst.snd)).prodMk continuous_snd have he : angularReparam (E := E) C ∘ (fun p : U => angularReparam g p.val)=Subtype.val := by funext p change ((p.val.1.1,C (g p.val.1.2)),p.val.2)=p.val rw [hinv _ (hU _ p.property)] apply Topology.IsEmbedding.of_comp hs hCs rw [he] exact Topology.IsEmbedding.subtypeVal omit [NormedAddCommGroup E] [NormedSpace ℝ E] in theorem angularReparam_injOn {g C : ℝ → ℝ} {J : Set ℝ} (hinv : ∀ T∈J,C (g T)=T) {U : Set (Plane × E)} (hU : ∀ p∈U,p.1.2∈J) : InjOn (angularReparam g) U := by intro p hp q hq he have hh := congrArg (angularReparam (E := E) C) he change ((p.1.1,C (g p.1.2)),p.2)=((q.1.1,C (g q.1.2)),q.2) at hh simpa only [hinv _ (hU _ hp),hinv _ (hU _ hq)] using hh end section open scoped ContDiff Topology BigOperators open Set Function theorem angular_packing_transport {m : ℕ} {ι : Type*} {I J : Set ℝ} (hI : IsOpen I) (hJ : IsOpen J) {A B : Plane → ℝ} (hA : ContDiffOn ℝ ∞ A (I ×ˢ J)) (hB : ContDiffOn ℝ ∞ B (I ×ˢ J)) {w g C : ℝ → ℝ} (hw : ContDiff ℝ ∞ w) (hg : ContDiffOn ℝ ∞ g (Ioo (0:ℝ) 1)) (hmap : MapsTo g (Ioo (0:ℝ) 1) J) (hC : Continuous C) (hinv : ∀ T∈Ioo (0:ℝ) 1,C (g T)=T) (hnorm : ∀ T∈Ioo (0:ℝ) 1,w (g T)*deriv g T=1) {G : ℝ × PlanePhase (Fin m) → ℝ} (hG : ContDiff ℝ ∞ G) (a b r : ι → ℝ) (ha : ∀ i,Ioo (a i) (b i)⊆I) {V : Set (Fin m → ℝ)} (U : ι → Set (Ambient (m+1))) (φ : ι → Ambient (m+1) → Plane × PlanePhase (Fin m)) (hU : ∀ i,IsOpen (U i)) (hBU : ∀ i,closedBall (m+1) (r i)⊆U i) (hφ : ∀ i,ContDiffOn ℝ ∞ (φ i) (U i)) (hemb : ∀ i,Topology.IsEmbedding (fun z : U i => φ i z)) (him : ∀ i,∀ z∈U i,φ i z∈(Ioo (a i) (b i) ×ˢ Ioo (0:ℝ) 1) ×ˢ (planeMoments ⁻¹' V)) (hform : ∀ i,∀ z∈U i,∀ v u, areaHamiltonianForm (normalizedAreaDensity A B g) G (φ i z) (fderiv ℝ (φ i) z v) (fderiv ℝ (φ i) z u)=standardForm v u) (hdis : Pairwise (fun i j => Disjoint (φ i '' closedBall (m+1) (r i)) (φ j '' closedBall (m+1) (r j)))) : ∃ ψ : ι → Ambient (m+1) → Plane × PlanePhase (Fin m), (∀ i,ContDiffOn ℝ ∞ (ψ i) (U i)) ∧ (∀ i,Topology.IsEmbedding (fun z : U i => ψ i z)) ∧ (∀ i,∀ z∈U i,ψ i z∈(Ioo (a i) (b i) ×ˢ J) ×ˢ (planeMoments ⁻¹' V)) ∧ (∀ i,∀ z∈U i,∀ v u, horizontalCoupling phaseArea (baseCoefficient A) (weightedSecondCoefficient B (fun q => w q.2) G) (ψ i z) (fderiv ℝ (ψ i) z v) (fderiv ℝ (ψ i) z u)=standardForm v u) ∧ Pairwise (fun i j => Disjoint (ψ i '' closedBall (m+1) (r i)) (ψ j '' closedBall (m+1) (r j))) := by let S : Set (Plane × PlanePhase (Fin m)) := (I ×ˢ Ioo (0:ℝ) 1) ×ˢ (Set.univ : Set (PlanePhase (Fin m))) have hS : IsOpen S := (hI.prod isOpen_Ioo).prod isOpen_univ have hmS : ∀ i,MapsTo (φ i) (U i) S := fun i z hz => ⟨⟨ha i (him i z hz).1.1,(him i z hz).1.2⟩,mem_univ _⟩ have hgs : ContDiffOn ℝ ∞ (angularReparam g) S := angularReparam_smoothOn hg (fun _ h => h.1.2) let ψ : ι → Ambient (m+1) → Plane × PlanePhase (Fin m) := fun i => angularReparam g ∘ φ i have hψ : ∀ i,ContDiffOn ℝ ∞ (ψ i) (U i) := fun i => hgs.comp (hφ i) (hmS i) have hCs : Continuous (angularReparam (E := PlanePhase (Fin m)) C) := (continuous_fst.fst.prodMk (hC.comp continuous_fst.snd)).prodMk continuous_snd refine ⟨ψ,hψ,?_,?_,?_,?_⟩ · intro i have he : angularReparam (E := PlanePhase (Fin m)) C ∘ (fun z : U i => ψ i z)= (fun z : U i => φ i z) := by funext z change (((φ i z).1.1,C (g (φ i z).1.2)),(φ i z).2)=φ i z rw [hinv _ (him i z z.property).1.2] apply Topology.IsEmbedding.of_comp (hψ i).continuousOn.domRestrict hCs change Topology.IsEmbedding (angularReparam C ∘ (fun z : U i => ψ i z)) rw [he] exact hemb i · intro i z hz exact ⟨⟨(him i z hz).1.1,hmap (him i z hz).1.2⟩,(him i z hz).2⟩ · intro i z hz v u have hp := hmS i hz have hq : ((φ i z).1.1,g (φ i z).1.2)∈I ×ˢ J := ⟨hp.1.1,hmap hp.1.2⟩ have hgd := (hg.contDiffAt (isOpen_Ioo.mem_nhds hp.1.2)).differentiableAt (by simp) have hψd : fderiv ℝ (ψ i) z= (fderiv ℝ (angularReparam g) (φ i z)).comp (fderiv ℝ (φ i) z) := fderiv_comp z ((hgs.contDiffAt (hS.mem_nhds hp)).differentiableAt (by simp)) (((hφ i).contDiffAt ((hU i).mem_nhds hz)).differentiableAt (by simp)) rw [hψd] simp only [ContinuousLinearMap.comp_apply] dsimp only [ψ,Function.comp_apply] rw [angularReparam_weighted_pullback phaseArea ((hA.contDiffAt ((hI.prod hJ).mem_nhds hq)).differentiableAt (by simp)) ((hB.contDiffAt ((hI.prod hJ).mem_nhds hq)).differentiableAt (by simp)) (hw.differentiable (by simp) _) hgd (hG.differentiable (by simp) _) (hnorm _ hp.1.2)] simpa only [areaHamiltonianForm_apply] using hform i z hz v u · intro i j hij rw [show ψ i=angularReparam g ∘ φ i from rfl,show ψ j=angularReparam g ∘ φ j from rfl, Set.image_comp,Set.image_comp] apply (hdis hij).image (angularReparam_injOn hinv (fun _ hp => hp.1.2) (U := S)) · rintro _ ⟨z,hz,rfl⟩ exact hmS i (hBU i hz) · rintro _ ⟨z,hz,rfl⟩ exact hmS j (hBU j hz) theorem finite_weighted_layer_packing {m : ℕ} {ι : Type*} {δ : ℝ} (hδ : 0<δ) (hsmall : δ<1/2) {I : Set ℝ} (hI : IsOpen I) (hc : Convex ℝ I) {A B : Plane → ℝ} (hA : ContDiffOn ℝ ∞ A (I ×ˢ Ioo (1/2-δ) (1/2+δ))) (hB : ContDiffOn ℝ ∞ B (I ×ˢ Ioo (1/2-δ) (1/2+δ))) (hbase : ∀ q∈I ×ˢ Ioo (1/2-δ) (1/2+δ),0 Disjoint (Ioo (a i) (b i)) (Ioo (a j) (b j)))) (hr' : ∀ i,0≤r' i) (hrr : ∀ i,r' i φ i z)) ∧ (∀ i,∀ z∈U i,φ i z∈(Ioo (a i) (b i) ×ˢ Ioo (1/2-δ) (1/2+δ)) ×ˢ (planeMoments ⁻¹' V)) ∧ (∀ i,∀ z∈U i,∀ v u, horizontalCoupling phaseArea (baseCoefficient A) (weightedSecondCoefficient B (fun q => positiveCircleClock δ (circleTurn q.2)) G) (φ i z) (fderiv ℝ (φ i) z v) (fderiv ℝ (φ i) z u)=standardForm v u) ∧ Pairwise (fun i j => Disjoint (φ i '' closedBall (m+1) (r' i)) (φ j '' closedBall (m+1) (r' j))) := by obtain ⟨g,hg,hm,_,hinv,hnorm,hℓ,hpos⟩ := exists_normalized_positive_area hδ hsmall hI hA hB hbase obtain ⟨U,φ,hU,hBU,hφ,hemb,him,hform,hdi⟩ := finite_area_layer_packing hI hc hℓ hpos F H ρ hF hH hρ hρpos hV hHp a b r r' ha hb hab hρa hρb hdis hr' hrr hsimplex hheight G he have hw : ContDiff ℝ ∞ (fun t => positiveCircleClock δ (circleTurn t)) := ((positiveCircleClock_smooth hδ hsmall).comp circleTurn_smooth).contDiff have hC : Continuous (fun t => intervalClock (-δ) δ (t-1/2)) := (intervalClock_smooth (-δ) δ).continuous.comp (continuous_id.sub continuous_const) have hasub : ∀ i,Ioo (a i) (b i)⊆I := by intro i have hseg : Icc (a i) (b i)⊆I := by simpa only [segment_eq_uIcc,uIcc_of_le (hab i).le] using hc.segment_subset (ha i) (hb i) exact Ioo_subset_Icc_self.trans hseg obtain ⟨ψ,hψ,heψ,hiψ,hfψ,hdψ⟩ := angular_packing_transport hI isOpen_Ioo hA hB hw hg hm hC hinv hnorm hG a b r' hasub U φ hU hBU hφ hemb him hform hdi exact ⟨U,ψ,hU,hBU,hψ,heψ,hiψ,hfψ,hdψ⟩ end section open scoped ContDiff open Set Function variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] variable {ι : Type*} [Fintype ι] def surfaceRemainder (K H : E → ℝ) (h : ι → E → ℝ) (ψ : E → E) (v : E) : ℝ := K (ψ v)-K v+H v-∑ i, h i v @[fun_prop] theorem surfaceRemainder_smooth {K H : E → ℝ} {h : ι → E → ℝ} {ψ : E → E} (hK : ContDiff ℝ ∞ K) (hH : ContDiff ℝ ∞ H) (hh : ∀ i, ContDiff ℝ ∞ (h i)) (hψ : ContDiff ℝ ∞ ψ) : ContDiff ℝ ∞ (surfaceRemainder K H h ψ) := by exact ((hK.comp hψ).sub hK |>.add hH).sub (ContDiff.sum (fun i _ => hh i)) omit [NormedAddCommGroup E] [NormedSpace ℝ E] in theorem surfaceRemainder_gap {K H H₀ C : E → ℝ} {h : ι → E → ℝ} {ψ : E → E} {g ε : ℝ} {v : E} (hgap : g ≤ K (ψ v)-K v+H₀ v-C v) (harea : |H v-H₀ v| < 2*ε) (hcap : (∑ i, h i v)-C v < ε) : g-3*ε < surfaceRemainder K H h ψ v := by have ha := (abs_lt.mp harea).1 dsimp [surfaceRemainder] linarith def surfaceFirstLayer (S : ℝ → ℝ) (H : E → ℝ) (ψ : E → E) (p : ℝ × E) : ℝ := S p.1*H (ψ p.2) def surfaceFinalLayer (J : E → ℝ) (h : ι → E → ℝ) (R : E → ℝ) (ρ : ι → ℝ → ℝ) (ρ₀ : ℝ → ℝ) (p : ℝ × E) : ℝ := J p.2+(∑ i, ρ i p.1*h i p.2)+ρ₀ p.1*R p.2 @[fun_prop] theorem surfaceFirstLayer_smooth {S : ℝ → ℝ} {H : E → ℝ} {ψ : E → E} (hS : ContDiff ℝ ∞ S) (hH : ContDiff ℝ ∞ H) (hψ : ContDiff ℝ ∞ ψ) : ContDiff ℝ ∞ (surfaceFirstLayer S H ψ) := (hS.comp contDiff_fst).mul ((hH.comp hψ).comp contDiff_snd) @[fun_prop] theorem surfaceFinalLayer_smooth {J R : E → ℝ} {h : ι → E → ℝ} {ρ : ι → ℝ → ℝ} {ρ₀ : ℝ → ℝ} (hJ : ContDiff ℝ ∞ J) (hR : ContDiff ℝ ∞ R) (hh : ∀ i, ContDiff ℝ ∞ (h i)) (hρ : ∀ i, ContDiff ℝ ∞ (ρ i)) (hρ₀ : ContDiff ℝ ∞ ρ₀) : ContDiff ℝ ∞ (surfaceFinalLayer J h R ρ ρ₀) := ((hJ.comp contDiff_snd).add (ContDiff.sum (fun i _ => (hρ i |>.comp contDiff_fst).mul (hh i |>.comp contDiff_snd)))).add ((hρ₀.comp contDiff_fst).mul (hR.comp contDiff_snd)) theorem first_partial_eq_deriv {F : ℝ × E → ℝ} {p : ℝ × E} (hF : DifferentiableAt ℝ F p) : fderiv ℝ F p (1,0)=deriv (fun s => F (s,p.2)) p.1 := by exact (hF.hasFDerivAt.comp_hasDerivAt p.1 ((hasDerivAt_id p.1).prodMk (hasDerivAt_const p.1 p.2))).deriv.symm theorem surfaceFirstLayer_firstDerivative {S : ℝ → ℝ} {H : E → ℝ} {ψ : E → E} (hS : ContDiff ℝ ∞ S) (hH : ContDiff ℝ ∞ H) (hψ : ContDiff ℝ ∞ ψ) (p : ℝ × E) : fderiv ℝ (surfaceFirstLayer S H ψ) p (1,0)=deriv S p.1*H (ψ p.2) := by rw [first_partial_eq_deriv ((surfaceFirstLayer_smooth hS hH hψ).differentiable (by simp) p)] simpa only [surfaceFirstLayer] using ((hS.differentiable (by simp) p.1).hasDerivAt.mul_const (H (ψ p.2))).deriv theorem surfaceFinalLayer_firstDerivative {J R : E → ℝ} {h : ι → E → ℝ} {ρ : ι → ℝ → ℝ} {ρ₀ : ℝ → ℝ} (hJ : ContDiff ℝ ∞ J) (hR : ContDiff ℝ ∞ R) (hh : ∀ i, ContDiff ℝ ∞ (h i)) (hρ : ∀ i, ContDiff ℝ ∞ (ρ i)) (hρ₀ : ContDiff ℝ ∞ ρ₀) (p : ℝ × E) : fderiv ℝ (surfaceFinalLayer J h R ρ ρ₀) p (1,0)= (∑ i, deriv (ρ i) p.1*h i p.2)+deriv ρ₀ p.1*R p.2 := by rw [first_partial_eq_deriv ((surfaceFinalLayer_smooth hJ hR hh hρ hρ₀).differentiable (by simp) p)] have hd := ((hasDerivAt_const p.1 (J p.2)).add (HasDerivAt.sum (u := Finset.univ) (fun i _ => (hρ i |>.differentiable (by simp) p.1).hasDerivAt.mul_const (h i p.2)))).add ((hρ₀.differentiable (by simp) p.1).hasDerivAt.mul_const (R p.2)) have he : (((fun _ : ℝ => J p.2) + ∑ i, fun s => ρ i s*h i p.2) + fun s => ρ₀ s*R p.2) = fun s => surfaceFinalLayer J h R ρ ρ₀ (s,p.2) := by funext s simp [surfaceFinalLayer] rw [he] at hd simpa only [zero_add] using hd.deriv theorem surfaceFinalLayer_firstDerivative_nonneg {J R : E → ℝ} {h : ι → E → ℝ} {ρ : ι → ℝ → ℝ} {ρ₀ : ℝ → ℝ} (hJ : ContDiff ℝ ∞ J) (hR : ContDiff ℝ ∞ R) (hh : ∀ i, ContDiff ℝ ∞ (h i)) (hρ : ∀ i, ContDiff ℝ ∞ (ρ i)) (hρ₀ : ContDiff ℝ ∞ ρ₀) (hm : ∀ i, Monotone (ρ i)) (hm₀ : Monotone ρ₀) (p : ℝ × E) (hpos : ∀ i, 0 ≤ h i p.2) (hRpos : 0 ≤ R p.2) : 0 ≤ fderiv ℝ (surfaceFinalLayer J h R ρ ρ₀) p (1,0) := by rw [surfaceFinalLayer_firstDerivative hJ hR hh hρ hρ₀] exact add_nonneg (Finset.sum_nonneg (fun i _ => mul_nonneg (hm i |>.deriv_nonneg) (hpos i))) (mul_nonneg hm₀.deriv_nonneg hRpos) omit [NormedAddCommGroup E] [NormedSpace ℝ E] in theorem surfaceFinalLayer_lower {J R : E → ℝ} {h : ι → E → ℝ} {ρ : ι → ℝ → ℝ} {ρ₀ : ℝ → ℝ} {s : ℝ} (hz : ∀ i, ρ i s=0) (hz₀ : ρ₀ s=0) (v : E) : surfaceFinalLayer J h R ρ ρ₀ (s,v)=J v := by simp [surfaceFinalLayer,hz,hz₀] omit [NormedAddCommGroup E] [NormedSpace ℝ E] in theorem surfaceFinalLayer_upper {K H : E → ℝ} {h : ι → E → ℝ} {ψ : E → E} {ρ : ι → ℝ → ℝ} {ρ₀ : ℝ → ℝ} {s : ℝ} (ho : ∀ i, ρ i s=1) (ho₀ : ρ₀ s=1) (v : E) : surfaceFinalLayer (K-H) h (surfaceRemainder K H h ψ) ρ ρ₀ (s,v)=K (ψ v) := by simp only [surfaceFinalLayer,surfaceRemainder,ho,ho₀,Pi.sub_apply,one_mul] ring end open scoped ContDiff open Set Function theorem surfaceFinalLayer_on_ordered_interval {N : ℕ} {E : Type*} (J R : E → ℝ) (h : Fin N → E → ℝ) {lo hi : Fin (N+1) → ℝ} {ρ : Fin (N+1) → ℝ → ℝ} (hord : ∀ i j, i < j → hi i < lo j) (hends : ∀ i, (∀ s, s ≤ lo i → ρ i s=0) ∧ (∀ s, hi i ≤ s → ρ i s=1)) (i : Fin N) {s : ℝ} (hs : s ∈ Icc (lo i.castSucc) (hi i.castSucc)) (v : E) : surfaceFinalLayer J h R (fun j => ρ j.castSucc) (ρ (Fin.last N)) (s,v)= J v+(∑ j ∈ Finset.univ.filter (fun j : Fin N => j < i), h j v)+ρ i.castSucc s*h i v := by classical have hlast : ρ (Fin.last N) s=0 := (hends _).1 s (hs.2.trans (hord _ _ (by simp)).le) have he (j : Fin N) : ρ j.castSucc s*h j v = (if j < i then h j v else 0)+(if j=i then ρ i.castSucc s*h i v else 0) := by rcases lt_trichotomy j i with hj | hj | hj · have hr : ρ j.castSucc s=1 := (hends _).2 s ((hord _ _ (by simpa using hj)).le.trans hs.1) simp [hj,ne_of_lt hj,hr] · subst j simp · have hr : ρ j.castSucc s=0 := (hends _).1 s (hs.2.trans (hord _ _ (by simpa using hj)).le) simp [not_lt.mpr hj.le,ne_of_gt hj,hr] unfold surfaceFinalLayer rw [hlast,zero_mul,add_zero] rw [Finset.sum_congr rfl (fun j (_ : j ∈ (Finset.univ : Finset (Fin N))) => he j)] simp only [Finset.sum_add_distrib] rw [← Finset.sum_filter] simp ring theorem exists_surface_layer_clocks (N : ℕ) {a b : ℝ} (hab : a < b) : ∃ lo hi : Fin (N+1) → ℝ, ∃ ρ : Fin (N+1) → ℝ → ℝ, (∀ i, a ρ j.castSucc) (ρ (Fin.last N)) (s,v)= J v+(∑ j ∈ Finset.univ.filter (fun j : Fin N => j < i), h j v)+ρ i.castSucc s*h i v) := by obtain ⟨lo,hi,ρ,hbd,hord,hsm,hpos,hcol⟩ := exists_ordered_smooth_clocks (N+1) hab refine ⟨lo,hi,ρ,hbd,hord,hsm,hpos,hcol,?_⟩ intro E J R h i s hs v apply surfaceFinalLayer_on_ordered_interval J R h hord _ i hs v intro j obtain ⟨δ,hδ,hz,ho⟩ := hcol j exact ⟨fun s hs => hz s (by linarith),fun s hs => ho s (by linarith)⟩ end PackingSufficiencySupport.Hamiltonian end end OAI