import Mathlib import OAI.Geometry.BallPacking.DiskMaps.CubicAreaExhaustion namespace OAI noncomputable section namespace PackingSufficiencySupport.Hamiltonian section open scoped ContDiff Topology Manifold open Set Function variable {P F : Type*} [NormedAddCommGroup P] [NormedSpace ℝ P] [FiniteDimensional ℝ P] [NormedAddCommGroup F] [NormedSpace ℝ F] theorem exists_smooth_convex_selection {Y : Set P} (hY : IsClosed Y) (S : P → Set F) (hopen : ∀ v, IsOpen {y | v ∈ S y}) (hconv : ∀ y ∈ Y, Convex ℝ (S y)) (hne : ∀ y ∈ Y, (S y).Nonempty) : ∃ c : P → F, ContDiff ℝ ∞ c ∧ ∀ y ∈ Y, c y ∈ S y := by classical choose v hv using fun y : Y => hne y y.property let U : Y → Set P := fun y => {x | v y ∈ S x} obtain ⟨ρ,hρ⟩ := SmoothPartitionOfUnity.exists_isSubordinate 𝓘(ℝ,P) hY U (fun y => hopen (v y)) (by intro y hy exact mem_iUnion_of_mem ⟨y,hy⟩ (hv ⟨y,hy⟩)) refine ⟨fun y => ∑ᶠ i, ρ i y • v i,?_,?_⟩ · apply contDiff_iff_contDiffAt.mpr intro y exact ρ.contDiffAt_finsum (fun _ _ => contDiffAt_const) · intro y hy apply ρ.finsum_smul_mem_convex hy _ (hconv y hy) intro i hi exact hρ i (subset_tsupport (ρ i) hi) end section open Set Function theorem exists_strict_interval_extension {a b l u L : ℝ} (hal : a < l) (hub : u < b) (hLu : u-l < L) (hLb : L < b-a) : ∃ c, a < c ∧ c+L < b ∧ c < l ∧ u < c+L := by have hh : max a (u-L) < min l (b-L) := by rw [max_lt_iff,lt_min_iff,lt_min_iff] constructor <;> constructor <;> linarith obtain ⟨c,hlo,hhi⟩ := exists_between hh exact ⟨c,(le_max_left _ _).trans_lt hlo, by linarith [(lt_of_lt_of_le hhi (min_le_right _ _))], hhi.trans_le (min_le_left _ _), by linarith [(lt_of_le_of_lt (le_max_right _ _) hlo)]⟩ theorem exists_strict_nested_intervals {n : ℕ} (a b L : Fin n → ℝ) (ha : Antitone a) (hb : Monotone b) (hL : StrictMono L) (hfit : ∀ i, L i < b i-a i) : ∃ c : Fin n → ℝ, (∀ i, a i < c i ∧ c i+L i < b i) ∧ ∀ ⦃i j⦄, i < j → c j < c i ∧ c i+L i < c j+L j := by induction n with | zero => exact ⟨fun i => i.elim0,by simp,by simp⟩ | succ n ih => cases n with | zero => refine ⟨fun i => (a i+b i-L i)/2,?_,?_⟩ · intro i; have := hfit i; constructor <;> linarith · intro i j hij have : i = j := by omega exact (lt_irrefl _ (this ▸ hij)).elim | succ m => obtain ⟨c,hc,hnest⟩ := ih (a ∘ Fin.castSucc) (b ∘ Fin.castSucc) (L ∘ Fin.castSucc) (ha.comp_monotone Fin.strictMono_castSucc.monotone) (hb.comp Fin.strictMono_castSucc.monotone) (hL.comp Fin.strictMono_castSucc) (fun i => hfit i.castSucc) let i₀ : Fin (m+1) := Fin.last m let j₀ : Fin (m+2) := Fin.last (m+1) have hij : i₀.castSucc < j₀ := Fin.castSucc_lt_last _ obtain ⟨d,hd⟩ := exists_strict_interval_extension ((ha hij.le).trans_lt (hc i₀).1) ((hc i₀).2.trans_le (hb hij.le)) (by simpa only [Function.comp_apply,add_sub_cancel_left] using hL hij) (hfit j₀) refine ⟨Fin.snoc c d,?_,?_⟩ · intro i refine Fin.lastCases ?_ (fun j => ?_) i · simpa only [Fin.snoc_last] using And.intro hd.1 hd.2.1 · simpa only [Fin.snoc_castSucc,Function.comp_apply] using hc j · intro i j hij' have hi : i ≠ Fin.last (m+1) := by intro hh subst i exact not_lt_of_ge (Fin.le_last _) hij' obtain ⟨i,rfl⟩ := Fin.eq_castSucc_of_ne_last hi revert hij' refine Fin.lastCases ?_ (fun j hj => ?_) j · intro _ simp only [Fin.snoc_castSucc,Fin.snoc_last] rcases lt_or_eq_of_le (Fin.le_last i) with hi | hi · have hh := hnest hi exact ⟨hd.2.2.1.trans hh.1,hh.2.trans hd.2.2.2⟩ · subst i exact hd.2.2 · simpa only [Fin.snoc_castSucc,Function.comp_apply] using hnest (show i < j from hj) end section open scoped ContDiff Topology open Set Function variable {n : ℕ} def StrictNestedStarts (L c : Fin n → ℝ) : Prop := ∀ ⦃i j : Fin n⦄, i < j → c j < c i ∧ c i+L i < c j+L j def IntervalPlacements (L : Fin n → ℝ) (V : Fin n → Set ℝ) : Set (Fin n → ℝ) := {c | (∀ i, c i ∈ V i ∧ c i+L i ∈ V i) ∧ StrictNestedStarts L c} theorem intervalPlacements_convex (L : Fin n → ℝ) (V : Fin n → Set ℝ) (hV : ∀ i, Convex ℝ (V i)) : Convex ℝ (IntervalPlacements L V) := by intro c hc d hd α β hα hβ hab refine ⟨?_,?_⟩ · intro i have he : (α • c+β • d) i+L i = α*(c i+L i)+β*(d i+L i) := by simp only [Pi.add_apply,Pi.smul_apply,smul_eq_mul] have he := congrArg (fun t : ℝ => t*L i) hab nlinarith only [he] constructor · exact hV i (hc.1 i).1 (hd.1 i).1 hα hβ hab · rw [he] exact hV i (hc.1 i).2 (hd.1 i).2 hα hβ hab · intro i j hij let A : (Fin n → ℝ) →ₗ[ℝ] ℝ := (LinearMap.proj j : (Fin n → ℝ) →ₗ[ℝ] ℝ) - LinearMap.proj i let B : (Fin n → ℝ) →ₗ[ℝ] ℝ := (LinearMap.proj i : (Fin n → ℝ) →ₗ[ℝ] ℝ) - LinearMap.proj j have ha : A c < 0 := sub_neg.mpr (hc.2 hij).1 have hb : A d < 0 := sub_neg.mpr (hd.2 hij).1 have ha' : B c < L j-L i := by change c i-c j < _; linarith [(hc.2 hij).2] have hb' : B d < L j-L i := by change d i-d j < _; linarith [(hd.2 hij).2] have hh := (convex_halfSpace_lt A.isLinear 0) ha hb hα hβ hab have hh' := (convex_halfSpace_lt B.isLinear (L j-L i)) ha' hb' hα hβ hab change (α • c+β • d) j-(α • c+β • d) i < 0 at hh change (α • c+β • d) i-(α • c+β • d) j < L j-L i at hh' exact ⟨sub_neg.mp hh,by linarith⟩ variable {P : Type*} [NormedAddCommGroup P] [NormedSpace ℝ P] [FiniteDimensional ℝ P] theorem exists_smooth_nested_intervals {Y : Set P} (hY : IsClosed Y) (L : Fin n → ℝ) (hL : StrictMono L) (V : P → Fin n → Set ℝ) (hopen : ∀ i t, IsOpen {y | t ∈ V y i}) (hconv : ∀ y ∈ Y, ∀ i, Convex ℝ (V y i)) (hroom : ∀ y ∈ Y, ∃ a b : Fin n → ℝ, Antitone a ∧ Monotone b ∧ (∀ i, L i < b i-a i) ∧ ∀ i, Ioo (a i) (b i) ⊆ V y i) (hpos : ∀ i, 0 < L i) : ∃ c : P → (Fin n → ℝ), ContDiff ℝ ∞ c ∧ ∀ y ∈ Y, (∀ i, Icc (c y i) (c y i+L i) ⊆ V y i) ∧ StrictNestedStarts L (c y) := by have hsopen (c : Fin n → ℝ) : IsOpen {y | c ∈ IntervalPlacements L (V y)} := by change IsOpen {y | (∀ i, c i ∈ V y i ∧ c i+L i ∈ V y i) ∧ StrictNestedStarts L c} by_cases hn : StrictNestedStarts L c · simp only [hn,and_true,ofPred_forall,ofPred_and] exact isOpen_iInter_of_finite (fun i => (hopen i (c i)).inter (hopen i (c i+L i))) · simp only [hn,and_false,ofPred_false,isOpen_empty] have hsne (y : P) (hy : y ∈ Y) : (IntervalPlacements L (V y)).Nonempty := by obtain ⟨a,b,ha,hb,hfit,hsub⟩ := hroom y hy obtain ⟨c,hc,hn⟩ := exists_strict_nested_intervals a b L ha hb hL hfit refine ⟨c,?_,hn⟩ intro i have hLi := hpos i exact ⟨hsub i ⟨(hc i).1,by linarith [(hc i).2]⟩, hsub i ⟨by linarith [(hc i).1],(hc i).2⟩⟩ obtain ⟨c,hc,hmem⟩ := exists_smooth_convex_selection hY (fun y => IntervalPlacements L (V y)) hsopen (fun y hy => intervalPlacements_convex L (V y) (hconv y hy)) hsne refine ⟨c,hc,?_⟩ intro y hy refine ⟨?_,(hmem y hy).2⟩ intro i exact (hconv y hy i).ordConnected.out ((hmem y hy).1 i).1 ((hmem y hy).1 i).2 end section open scoped ContDiff Topology open Set Function variable {P : Type*} [TopologicalSpace P] def RoundedMargins {n : ℕ} (C : P → Fin n → ℝ) (L w : Fin n → ℝ) (V : P → Fin n → Set ℝ) (q s : ℝ) (y : P) : Prop := 0 < s ∧ (∀ i, C y i-s*L i/2 ∈ V y i ∧ C y i+s*L i/2 ∈ V y i) ∧ ∀ ⦃i j⦄, i < j → C y j-q*s*L j/2 < C y i-s*L i/2 ∧ C y i+s*L i/2 < C y j+q*s*L j/2 ∧ w i < q*w j theorem roundedMargins_open {n : ℕ} (C : P → Fin n → ℝ) (hC : Continuous C) (L w : Fin n → ℝ) (V : P → Fin n → Set ℝ) (hV : ∀ i, IsOpen {p : P × ℝ | p.2 ∈ V p.1 i}) : IsOpen {p : Plane × P | RoundedMargins C L w V p.1.1 p.1.2 p.2} := by have hlo (i : Fin n) : IsOpen {p : Plane × P | C p.2 i-p.1.2*L i/2 ∈ V p.2 i} := (hV i).preimage (continuous_snd.prodMk (((continuous_apply i).comp (hC.comp continuous_snd)).sub ((continuous_fst.snd.mul continuous_const).div_const 2))) have hhi (i : Fin n) : IsOpen {p : Plane × P | C p.2 i+p.1.2*L i/2 ∈ V p.2 i} := (hV i).preimage (continuous_snd.prodMk (((continuous_apply i).comp (hC.comp continuous_snd)).add ((continuous_fst.snd.mul continuous_const).div_const 2))) have hpair (i j : Fin n) : IsOpen {p : Plane × P | i < j → C p.2 j-p.1.1*p.1.2*L j/2 < C p.2 i-p.1.2*L i/2 ∧ C p.2 i+p.1.2*L i/2 < C p.2 j+p.1.1*p.1.2*L j/2 ∧ w i < p.1.1*w j} := by by_cases hij : i < j · simp only [hij,true_implies,ofPred_and] exact (isOpen_lt (by fun_prop) (by fun_prop)).inter ((isOpen_lt (by fun_prop) (by fun_prop)).inter (isOpen_lt (by fun_prop) (by fun_prop))) · simp only [hij,false_implies,ofPred_true,isOpen_univ] change IsOpen {p : Plane × P | 0 < p.1.2 ∧ (∀ i, C p.2 i-p.1.2*L i/2 ∈ V p.2 i ∧ C p.2 i+p.1.2*L i/2 ∈ V p.2 i) ∧ ∀ i j, i < j → _} simp only [ofPred_and,ofPred_forall] exact (isOpen_lt continuous_const continuous_fst.snd).inter ((isOpen_iInter_of_finite (fun i => (hlo i).inter (hhi i))).inter (isOpen_iInter_of_finite (fun i => isOpen_iInter_of_finite (fun j => by simpa only [ofPred_forall,ofPred_and] using hpair i j)))) theorem exists_exact_area_rounded_margins {n : ℕ} {Y : Set P} (hY : IsCompact Y) (c : P → Fin n → ℝ) (hc : Continuous c) (L w : Fin n → ℝ) (hw : StrictMono w) (V : P → Fin n → Set ℝ) (hV : ∀ i, IsOpen {p : P × ℝ | p.2 ∈ V p.1 i}) (hplace : ∀ y ∈ Y, (∀ i, c y i ∈ V y i ∧ c y i+L i ∈ V y i) ∧ StrictNestedStarts L (c y)) : ∃ q k : ℝ, 0 < q ∧ q < 1 ∧ 0 < k ∧ Icc (-q) q ×ˢ Icc (-q) q ⊆ squeezedDisk k ∧ 0 < squeezedArea k ∧ ∀ y ∈ Y, RoundedMargins (fun y i => c y i+L i/2) L w V q (4/squeezedArea k) y := by let C : P → Fin n → ℝ := fun y i => c y i+L i/2 have hC : Continuous C := by apply continuous_pi intro i exact ((continuous_apply i).comp hc).add continuous_const apply exists_rounded_square_with_control hY continuous_id (roundedMargins_open C hC L w V hV) intro y hy change RoundedMargins C L w V 1 1 y refine ⟨by norm_num,?_,?_⟩ · intro i simpa only [C,one_mul,add_sub_cancel_right,add_assoc,add_halves] using (hplace y hy).1 i · intro i j hij dsimp [C] simp only [one_mul,add_sub_cancel_right,add_assoc,add_halves] exact ⟨((hplace y hy).2 hij).1,((hplace y hy).2 hij).2,hw hij⟩ end section open scoped ContDiff Topology BigOperators open Set Function theorem exists_thin_rectangle_placement {m : ℕ} (h : Fin m → ℝ) (hh : ∀ i, 0 < h i) (hmono : StrictMono h) {δ : ℝ} (hδ : 0 < δ) : ∃ (L w c : Fin m → ℝ), (∀ i, 0 < L i) ∧ (∀ i, 0 < w i ∧ w i < 1) ∧ StrictMono w ∧ (∀ i, L i*w i = h i) ∧ StrictNestedStarts L c ∧ ∀ i, c i ∈ Ioo 0 (h i+δ/2) ∧ c i+L i ∈ Ioo 0 (h i+δ/2) := by let B : ℝ := 1+∑ i, h i have hB : 1 ≤ B := by have hs : 0 ≤ ∑ i, h i := Finset.sum_nonneg (fun i _ => (hh i).le) dsimp [B]; linarith have hiB (i : Fin m) : h i < B := by have hi := Finset.single_le_sum (s := Finset.univ) (fun j _ => (hh j).le) (Finset.mem_univ i) dsimp [B]; linarith let ε := min (δ/8) (1/4) have hε : 0 < ε := lt_min (by positivity) (by norm_num) have hεδ : ε ≤ δ/8 := min_le_left _ _ have hε1 : ε ≤ 1/4 := min_le_right _ _ let β := ε/(2*B) have hd : 0 < 2*B := by linarith have hβ : 0 < β := div_pos hε hd have hβ1 : β < 1 := by apply (div_lt_one hd).mpr; linarith have hB0 : B ≠ 0 := ne_of_gt (by linarith) have hβB : β*B = ε/2 := by dsimp [β]; field_simp [hB0] have hsmall (i : Fin m) : β*h i < ε := by have he := mul_lt_mul_of_pos_left (hiB i) hβ rw [hβB] at he linarith let L i := h i+ε let w i := h i/(h i+ε) let c i := ε-β*h i have hL i : 0 < L i := add_pos (hh i) hε have hw i : 0 < w i ∧ w i < 1 := ⟨div_pos (hh i) (hL i),(div_lt_one (hL i)).mpr (by dsimp [L]; linarith)⟩ refine ⟨L,w,c,hL,hw,?_,?_,?_,?_⟩ · intro i j hij apply (div_lt_div_iff₀ (hL i) (hL j)).mpr dsimp [L] have he := mul_lt_mul_of_pos_right (hmono hij) hε nlinarith · intro i dsimp [L,w] field_simp [show h i+ε ≠ 0 from (hL i).ne'] · intro i j hij dsimp [c,L] have he := hmono hij have he1 := mul_lt_mul_of_pos_left he hβ have he2 := mul_lt_mul_of_pos_left he (sub_pos.mpr hβ1) constructor <;> nlinarith · intro i have hc : 0 < c i := sub_pos.mpr (hsmall i) have hcε : c i < ε := by dsimp [c]; nlinarith [mul_pos hβ (hh i)] have hlower : 0 < c i+L i := add_pos hc (hL i) have hupper : c i+L i < h i+δ/2 := by dsimp [L]; linarith exact ⟨⟨hc,(lt_add_of_pos_right (c i) (hL i)).trans hupper⟩,hlower,hupper⟩ end section open scoped ContDiff Topology open Set Function MeasureTheory def box (c : Plane) (a b : ℝ) : Set Plane := Icc (c.1-a) (c.1+a) ×ˢ Icc (c.2-b) (c.2+b) def openBox (c : Plane) (a b : ℝ) : Set Plane := Ioo (c.1-a) (c.1+a) ×ˢ Ioo (c.2-b) (c.2+b) def roundedMap (k : ℝ) (c : Plane) (a b : ℝ) (x : Plane) : Plane := affineRectangle c a b (squareSqueeze k x) def roundedBox (k : ℝ) (c : Plane) (a b : ℝ) : Set Plane := roundedMap k c a b '' closedRoundDisk 1 theorem affineRectangle_injective {a b : ℝ} (ha : a ≠ 0) (hb : b ≠ 0) (c : Plane) : Injective (affineRectangle c a b) := by intro x y he have h1 := congrArg Prod.fst he have h2 := congrArg Prod.snd he exact Prod.ext (mul_left_cancel₀ ha (add_left_cancel h1)) (mul_left_cancel₀ hb (add_left_cancel h2)) theorem affineRectangle_smooth (c : Plane) (a b : ℝ) : ContDiff ℝ ∞ (affineRectangle c a b) := contDiff_const.add (diagonalScale a b).contDiff theorem roundedMap_smooth (k : ℝ) (c : Plane) (a b : ℝ) : ContDiff ℝ ∞ (roundedMap k c a b) := (affineRectangle_smooth c a b).comp (squareSqueeze_smooth k) theorem roundedMap_injective {k a b : ℝ} (hk : k ≠ 0) (ha : a ≠ 0) (hb : b ≠ 0) (c : Plane) : Injective (roundedMap k c a b) := (affineRectangle_injective ha hb c).comp (squareSqueeze_injective hk) theorem affineRectangle_fderiv (c : Plane) (a b : ℝ) (x : Plane) : fderiv ℝ (affineRectangle c a b) x = diagonalScale a b := by exact ((diagonalScale a b).hasFDerivAt.const_add c).fderiv theorem roundedMap_det {k a b : ℝ} (c : Plane) (x : Plane) : (fderiv ℝ (roundedMap k c a b) x).det = a*b*(fderiv ℝ (squareSqueeze k) x).det := by change (fderiv ℝ (affineRectangle c a b ∘ squareSqueeze k) x).det = _ rw [fderiv_comp x ((affineRectangle_smooth c a b).differentiable (by simp) _) ((squareSqueeze_smooth k).differentiable (by simp) _),affineRectangle_fderiv, ContinuousLinearMap.det,ContinuousLinearMap.toLinearMap_comp,LinearMap.det_comp] change (diagonalScale a b).det * _ = _ rw [diagonalScale_det] theorem roundedMap_det_pos {k a b : ℝ} (hk : 0 < k) (ha : 0 < a) (hb : 0 < b) (c : Plane) (x : Plane) : 0 < (fderiv ℝ (roundedMap k c a b) x).det := by rw [roundedMap_det] exact mul_pos (mul_pos ha hb) (squareSqueeze_det_pos hk x) theorem roundedMap_mem_openBox {a b : ℝ} (ha : 0 < a) (hb : 0 < b) (k : ℝ) (c : Plane) (x : Plane) : roundedMap k c a b x ∈ openBox c a b := by have hx1 := Real.neg_one_lt_tanh (k*x.1) have hx2 := Real.tanh_lt_one (k*x.1) have hy1 := Real.neg_one_lt_tanh (k*x.2) have hy2 := Real.tanh_lt_one (k*x.2) change (c.1-a < c.1+a*Real.tanh (k*x.1) ∧ c.1+a*Real.tanh (k*x.1) < c.1+a) ∧ (c.2-b < c.2+b*Real.tanh (k*x.2) ∧ c.2+b*Real.tanh (k*x.2) < c.2+b) constructor <;> constructor <;> nlinarith theorem box_subset_roundedBox {q k a b : ℝ} (ha : 0 < a) (hb : 0 < b) (hs : Icc (-q) q ×ˢ Icc (-q) q ⊆ squeezedDisk k) (c : Plane) : box c (q*a) (q*b) ⊆ roundedBox k c a b := by intro x hx let z : Plane := ((x.1-c.1)/a,(x.2-c.2)/b) have hz : z ∈ Icc (-q) q ×ˢ Icc (-q) q := by change (-q ≤ (x.1-c.1)/a ∧ (x.1-c.1)/a ≤ q) ∧ (-q ≤ (x.2-c.2)/b ∧ (x.2-c.2)/b ≤ q) constructor <;> constructor · apply (le_div_iff₀ ha).mpr; linarith [hx.1.1] · apply (div_le_iff₀ ha).mpr; linarith [hx.1.2] · apply (le_div_iff₀ hb).mpr; linarith [hx.2.1] · apply (div_le_iff₀ hb).mpr; linarith [hx.2.2] obtain ⟨u,hu,he⟩ := hs hz refine ⟨u,hu,?_⟩ change affineRectangle c a b (squareSqueeze k u) = x rw [he] ext <;> simp [affineRectangle,z,mul_div_cancel₀ _ ha.ne',mul_div_cancel₀ _ hb.ne'] theorem openBox_subset_interior_roundedBox {q k a b : ℝ} (ha : 0 < a) (hb : 0 < b) (hs : Icc (-q) q ×ˢ Icc (-q) q ⊆ squeezedDisk k) (c : Plane) : openBox c (q*a) (q*b) ⊆ interior (roundedBox k c a b) := by apply (isOpen_Ioo.prod isOpen_Ioo).subset_interior_iff.mpr exact (prod_mono Ioo_subset_Icc_self Ioo_subset_Icc_self).trans (box_subset_roundedBox ha hb hs c) theorem roundedBox_compact (k : ℝ) (c : Plane) (a b : ℝ) : IsCompact (roundedBox k c a b) := (closedRoundDisk_isCompact 1).image (roundedMap_smooth k c a b).continuous theorem roundedBox_volume {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (k : ℝ) (c : Plane) : (volume (roundedBox k c a b)).toReal = a*b*squeezedArea k := by change (volume ((affineRectangle c a b ∘ squareSqueeze k) '' closedRoundDisk 1)).toReal = _ rw [image_comp] exact affineRectangle_squeezedArea ha hb c k end section open scoped ContDiff Topology open Set Function Filter variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] variable [FiniteDimensional ℝ E] theorem isInvertible_of_det_ne_zero {A : E →L[ℝ] E} (hA : A.det ≠ 0) : A.IsInvertible := ⟨(A.toLinearMap.equivOfDetNeZero hA).toContinuousLinearEquiv,rfl⟩ theorem smooth_map_nhds_eq {f : E → E} {x : E} (hf : ContDiffAt ℝ ∞ f x) (hd : (fderiv ℝ f x).det ≠ 0) : map f (𝓝 x) = 𝓝 (f x) := by obtain ⟨e,he⟩ := isInvertible_of_det_ne_zero hd have hs := hf.hasStrictFDerivAt (by simp) rw [← he] at hs exact hs.map_nhds_eq_of_equiv theorem image_interior_subset_of_det_ne_zero {f : E → E} {S : Set E} (hf : ∀ x ∈ interior S, ContDiffAt ℝ ∞ f x) (hd : ∀ x ∈ interior S, (fderiv ℝ f x).det ≠ 0) : f '' interior S ⊆ interior (f '' S) := by rintro _ ⟨x,hx,rfl⟩ apply mem_interior_iff_mem_nhds.mpr have hh := Filter.image_mem_map (m := f) (mem_interior_iff_mem_nhds.mp hx) rwa [smooth_map_nhds_eq (hf x hx) (hd x hx)] at hh variable [FiniteDimensional ℝ F] omit [FiniteDimensional ℝ E] [FiniteDimensional ℝ F] in theorem det_fderiv_conjugate (e : E ≃L[ℝ] F) {f : F → F} {x : E} (hf : DifferentiableAt ℝ f (e x)) : (fderiv ℝ (fun x => e.symm (f (e x))) x).det = (fderiv ℝ f (e x)).det := by have hd := e.symm.hasFDerivAt.comp x (hf.hasFDerivAt.comp x e.hasFDerivAt) change (fderiv ℝ (e.symm ∘ f ∘ e) x).det = _ rw [hd.fderiv] change LinearMap.det (e.symm.toLinearEquiv.toLinearMap ∘ₗ (fderiv ℝ f (e x)).toLinearMap ∘ₗ e.toLinearEquiv.toLinearMap) = _ exact LinearMap.det_conj _ e.symm.toLinearEquiv end open scoped ContDiff Topology open Set Function MeasureTheory def FitsPolar (c : Plane) (a b : ℝ) : Prop := 0 < c.1-a ∧ c.1+a < 1 ∧ 0 < c.2-b ∧ c.2+b < 1 theorem openBox_subset_polar {c : Plane} {a b : ℝ} (hfit : FitsPolar c a b) : openBox c a b ⊆ areaPolarDomain := by intro x hx exact ⟨hfit.1.trans hx.1.1,hfit.2.2.1.trans hx.2.1,hx.2.2.trans hfit.2.2.2⟩ def polarRoundedMap (k : ℝ) (c : Plane) (a b : ℝ) (x : Plane) : Plane := areaPolar (roundedMap k c a b x) def polarRoundedDisk (k : ℝ) (c : Plane) (a b : ℝ) : Set Plane := polarRoundedMap k c a b '' closedRoundDisk 1 theorem polarRoundedMap_injective {k a b : ℝ} (hk : 0 < k) (ha : 0 < a) (hb : 0 < b) (c : Plane) (hfit : FitsPolar c a b) : Injective (polarRoundedMap k c a b) := by intro x y he apply roundedMap_injective hk.ne' ha.ne' hb.ne' c exact areaPolar_injOn (openBox_subset_polar hfit (roundedMap_mem_openBox ha hb k c x)) (openBox_subset_polar hfit (roundedMap_mem_openBox ha hb k c y)) he theorem polarRoundedMap_smooth {a b : ℝ} (ha : 0 < a) (hb : 0 < b) (k : ℝ) (c : Plane) (hfit : FitsPolar c a b) : ContDiff ℝ ∞ (polarRoundedMap k c a b) := by apply contDiff_iff_contDiffAt.mpr intro x exact (areaPolar_smoothAt (openBox_subset_polar hfit (roundedMap_mem_openBox ha hb k c x)).1).comp x (roundedMap_smooth k c a b).contDiffAt theorem polarRoundedMap_det {a b : ℝ} (ha : 0 < a) (hb : 0 < b) (k : ℝ) (c : Plane) (hfit : FitsPolar c a b) (x : Plane) : (fderiv ℝ (polarRoundedMap k c a b) x).det = (fderiv ℝ (roundedMap k c a b) x).det := by have hp := (openBox_subset_polar hfit (roundedMap_mem_openBox ha hb k c x)).1 change (fderiv ℝ (areaPolar ∘ roundedMap k c a b) x).det = _ rw [fderiv_comp x ((areaPolar_smoothAt hp).differentiableAt (by simp)) ((roundedMap_smooth k c a b).differentiable (by simp) _), ContinuousLinearMap.det,ContinuousLinearMap.toLinearMap_comp,LinearMap.det_comp] change (fderiv ℝ areaPolar (roundedMap k c a b x)).det * _ = _ rw [areaPolar_det hp,one_mul] theorem polarRoundedMap_det_pos {k a b : ℝ} (hk : 0 < k) (ha : 0 < a) (hb : 0 < b) (c : Plane) (hfit : FitsPolar c a b) (x : Plane) : 0 < (fderiv ℝ (polarRoundedMap k c a b) x).det := by rw [polarRoundedMap_det ha hb k c hfit] exact roundedMap_det_pos hk ha hb c x theorem polarRoundedMap_capacity {a b : ℝ} (ha : 0 < a) (hb : 0 < b) (k : ℝ) (c : Plane) (hfit : FitsPolar c a b) (x : Plane) : Real.pi * radiusSq (polarRoundedMap k c a b x) = (roundedMap k c a b x).1 := areaPolar_capacity (openBox_subset_polar hfit (roundedMap_mem_openBox ha hb k c x)).1.le theorem polarRoundedDisk_volume {a b : ℝ} (ha : 0 < a) (hb : 0 < b) (k : ℝ) (c : Plane) (hfit : FitsPolar c a b) : (volume (polarRoundedDisk k c a b)).toReal = a*b*squeezedArea k := by change (volume ((areaPolar ∘ roundedMap k c a b) '' closedRoundDisk 1)).toReal = _ rw [image_comp] change (volume (areaPolar '' roundedBox k c a b)).toReal = _ rw [areaPolar_image_volume (roundedBox_compact k c a b).measurableSet (by rintro z ⟨x,_,rfl⟩ exact openBox_subset_polar hfit (roundedMap_mem_openBox ha hb k c x))] exact roundedBox_volume ha.le hb.le k c theorem polarRoundedDisk_nested {k a b a' b' : ℝ} {c c' : Plane} (ha' : 0 < a') (hb' : 0 < b') (hfit' : FitsPolar c' a' b') (hnest : roundedBox k c a b ⊆ interior (roundedBox k c' a' b')) : polarRoundedDisk k c a b ⊆ interior (polarRoundedDisk k c' a' b') := by change (areaPolar ∘ roundedMap k c a b) '' closedRoundDisk 1 ⊆ interior ((areaPolar ∘ roundedMap k c' a' b') '' closedRoundDisk 1) rw [image_comp,image_comp] apply (image_mono hnest).trans apply image_interior_subset_of_det_ne_zero · intro x hx obtain ⟨z,_,rfl⟩ := interior_subset hx exact areaPolar_smoothAt (openBox_subset_polar hfit' (roundedMap_mem_openBox ha' hb' k c' z)).1 · intro x hx obtain ⟨z,_,rfl⟩ := interior_subset hx rw [areaPolar_det (openBox_subset_polar hfit' (roundedMap_mem_openBox ha' hb' k c' z)).1] norm_num variable {P : Type*} [NormedAddCommGroup P] [NormedSpace ℝ P] theorem polarRoundedFamily_smoothOn {A : Set P} (_hA : IsOpen A) (C : P → Plane) (hC : ContDiff ℝ ∞ C) {a b : ℝ} (ha : 0 < a) (hb : 0 < b) (k : ℝ) (hfit : ∀ y ∈ A, FitsPolar (C y) a b) : ContDiffOn ℝ ∞ (fun p : P × Plane => polarRoundedMap k (C p.1) a b p.2) (A ×ˢ univ) := by intro p hp apply ContDiffAt.contDiffWithinAt have hg : ContDiff ℝ ∞ (fun p : P × Plane => roundedMap k (C p.1) a b p.2) := (hC.comp contDiff_fst).add ((diagonalScale a b).contDiff.comp ((squareSqueeze_smooth k).comp contDiff_snd)) exact (areaPolar_smoothAt (openBox_subset_polar (hfit p.1 hp.1) (roundedMap_mem_openBox ha hb k (C p.1) p.2)).1).comp p hg.contDiffAt end PackingSufficiencySupport.Hamiltonian end end OAI