import Mathlib
import OAI.Geometry.BallPacking.Degree.ChartFormZeroExtension
namespace OAI
noncomputable section
namespace PackingSufficiencySupport.FiniteMoment
open scoped BigOperators Topology ContDiff
open Set Filter Function
section
variable {ι E : Type*} [Fintype ι] [Nonempty ι]
[NormedAddCommGroup E] [InnerProductSpace ℝ E]
def partition (w : ι → E) (a : ι → ℝ) (x : E) : ℝ :=
∑ i, a i * Real.exp (inner ℝ (w i) x)
def gradient (w : ι → E) (a : ι → ℝ) (x : E) : E :=
∑ i, (a i * Real.exp (inner ℝ (w i) x)) • w i
def moment (w : ι → E) (a : ι → ℝ) (x : E) : E :=
(partition w a x)⁻¹ • gradient w a x
def Surrounds (w : ι → E) (p : E) : Prop :=
∃ δ : ℝ, 0<δ ∧ ∀ v : E, ∃ i, δ*‖v‖≤ inner ℝ (w i-p) v
theorem partition_pos {w : ι → E} {a : ι → ℝ} (ha : ∀ i,0
(((innerSL ℝ (w i)).hasFDerivAt (x := x)).exp).const_mul (a i))
convert! h using 1
· ext y
simp [partition]
· simp only [gradient,map_sum,map_smul,smul_smul,innerSL_apply_apply]
theorem partition_coercive [ProperSpace E] {w : ι → E} {a : ι → ℝ}
(ha : ∀ i,0 (mul_pos (ha k) (Real.exp_pos (inner ℝ (w k) x))).le) (Finset.mem_univ i)
exact tendsto_atTop_mono hb ((Real.tendsto_exp_atTop.comp
(tendsto_norm_cocompact_atTop.const_mul_atTop hδ)).const_mul_atTop (ha j))
theorem exists_gradient_zero [ProperSpace E] {w : ι → E} {a : ι → ℝ}
(ha : ∀ i,0 L (gradient w a x)) hz
change inner ℝ (gradient w a x) (gradient w a x)=0 at he
exact inner_self_eq_zero.mp he
theorem exp_difference_nonneg (u v : ℝ) :
0≤(Real.exp u-Real.exp v)*(u-v) := by
rcases le_total v u with h|h
· exact mul_nonneg (sub_nonneg.mpr (Real.exp_le_exp.mpr h)) (sub_nonneg.mpr h)
· exact mul_nonneg_of_nonpos_of_nonpos (sub_nonpos.mpr (Real.exp_le_exp.mpr h))
(sub_nonpos.mpr h)
theorem exp_difference_pos {u v : ℝ} (h : v w i-p) a x = Real.exp (-inner ℝ p x) •
(gradient w a x-partition w a x • p) := by
have he (i : ι) : (a i * Real.exp (inner ℝ (w i-p) x)) • (w i-p) =
Real.exp (-inner ℝ p x) • ((a i*Real.exp (inner ℝ (w i) x)) • w i-
(a i*Real.exp (inner ℝ (w i) x)) • p) := by
rw [inner_sub_left,sub_eq_add_neg (inner ℝ (w i) x),Real.exp_add]
simp only [smul_sub,smul_smul]
congr 1 <;> congr 1 <;> ring
simp only [gradient,he,←Finset.smul_sum,Finset.sum_sub_distrib,←Finset.sum_smul,partition]
theorem moment_eq_iff_gradient_center_zero {w : ι → E} {a : ι → ℝ}
(ha : ∀ i,0 w i-p) a x=0 := by
rw [gradient_center,smul_eq_zero]
simp only [Real.exp_ne_zero,false_or,sub_eq_zero]
dsimp [moment]
rw [inv_smul_eq_iff₀ (partition_pos ha x).ne']
theorem exists_unique_moment [ProperSpace E] {w : ι → E} {a : ι → ℝ}
(ha : ∀ i,0
((((innerSL ℝ (w i)).hasFDerivAt (x := x)).exp).const_mul (a i)).smul_const (w i))
convert! h using 1
· ext y
simp [gradient]
· simp only [hessian,smul_smul,innerSL_apply_apply]
theorem hessian_inner (w : ι → E) (a : ι → ℝ) (x v : E) :
inner ℝ (hessian w a x v) v=
∑ i,(a i*Real.exp (inner ℝ (w i) x))*(inner ℝ (w i) v)^2 := by
rw [hessian_apply,sum_inner]
apply Finset.sum_congr rfl
intro i _
rw [real_inner_smul_left]
ring
theorem hessian_positive {w : ι → E} {a : ι → ℝ} (ha : ∀ i,0 gradient (fun i => w i-q.1.2) q.1.1 q.2) := by
apply ContDiff.sum
intro i _
have ha : ContDiff ℝ ∞ (fun q : ((ι → ℝ) × E) × E => q.1.1 i) := by fun_prop
have hu : ContDiff ℝ ∞ (fun q : ((ι → ℝ) × E) × E => w i-q.1.2) := by fun_prop
exact (ha.mul ((hu.inner ℝ contDiff_snd).exp)).smul hu
end
variable {ι E : Type*} [Fintype ι] [Nonempty ι]
[NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
def inverse (w : ι → E) (a : ι → ℝ) (p : E) : E := invFun (moment w a) p
theorem inverse_spec {w : ι → E} {a : ι → ℝ} (ha : ∀ i,0 isOpen_lt continuous_const (continuous_apply i))
convert h using 1
ext a
simp
theorem inverse_contDiffAt {w : ι → E} {a : ι → ℝ} (ha : ∀ i,0 inverse w q.1 q.2) (a,p) := by
let q := (a,p)
let x := inverse w a p
let F : ((ι → ℝ) × E) × E → E :=
fun t => gradient (fun i => w i-t.1.2) t.1.1 t.2
have hF : ContDiff ℝ ∞ F := centeredGradient_contDiff w
have hpartial : (fderiv ℝ F (q,x)).comp (ContinuousLinearMap.inr ℝ ((ι → ℝ) × E) E)=
hessian (fun i => w i-p) a x := by
have hc := ((hF.differentiable (by simp) (q,x)).hasFDerivAt.comp x
((hasFDerivAt_const q x).prodMk (hasFDerivAt_id x)))
exact hc.unique (gradient_hasFDerivAt (fun i => w i-p) a x)
have hinv : (fderiv ℝ F (q,x) ∘L ContinuousLinearMap.inr ℝ ((ι → ℝ) × E) E).IsInvertible := by
rw [hpartial]
exact hessian_isInvertible ha (fun _ hv => surrounds_separates hp hv) x
have hzero : F (q,x)=0 :=
(moment_eq_iff_gradient_center_zero ha p x).mp (inverse_spec ha hp)
have hapos : ∀ᶠ t : (ι → ℝ) × E in 𝓝 q,∀ i,0 inverse w q.1 q.2)
{q | (∀ i,0 (inverse_contDiffAt hq.1 hq.2).contDiffWithinAt
end PackingSufficiencySupport.FiniteMoment
namespace PackingSufficiencySupport.FiniteMoment.Radial
open scoped BigOperators Topology ContDiff
open Set Filter Function
abbrev Plane := ℝ × ℝ
variable {ι : Type*} [Fintype ι]
def monomial (k : ℕ × ℕ) (r : Plane) : ℝ := r.1^k.1 * r.2^k.2
def polynomial (k : ι → ℕ × ℕ) (a : ι → ℝ) (r : Plane) : ℝ :=
∑ i, a i * monomial (k i) r
def numerator (k : ι → ℕ × ℕ) (a : ι → ℝ) (r : Plane) : Plane :=
(∑ i, ((k i).1:ℝ)*a i*monomial (k i) r,
∑ i, ((k i).2:ℝ)*a i*monomial (k i) r)
def moment (k : ι → ℕ × ℕ) (a : ι → ℝ) (r : Plane) : Plane :=
(polynomial k a r)⁻¹ • numerator k a r
def centered (k : ι → ℕ × ℕ) (a : ι → ℝ) (p r : Plane) : Plane :=
(∑ i, (((k i).1:ℝ)-p.1)*a i*monomial (k i) r,
∑ i, (((k i).2:ℝ)-p.2)*a i*monomial (k i) r)
theorem monomial_nonneg (k : ℕ × ℕ) {r : Plane} (hr : 0≤r.1 ∧ 0≤r.2) :
0 ≤ monomial k r := mul_nonneg (pow_nonneg hr.1 _) (pow_nonneg hr.2 _)
theorem polynomial_pos {k : ι → ℕ × ℕ} {a : ι → ℝ} (ha : ∀ i,0 polynomial k q.1 q.2) := by
apply ContDiff.sum
intro i _
dsimp [monomial]
fun_prop
theorem centered_contDiff (k : ι → ℕ × ℕ) :
ContDiff ℝ ∞ (fun q : ((ι → ℝ) × Plane) × Plane => centered k q.1.1 q.1.2 q.2) := by
apply ContDiff.prodMk <;> apply ContDiff.sum <;> intro i _ <;> dsimp [monomial] <;> fun_prop
theorem centered_eq (k : ι → ℕ × ℕ) (a : ι → ℝ) (p r : Plane) :
centered k a p r=numerator k a r-polynomial k a r • p := by
apply Prod.ext
· change (∑ i, (((k i).1:ℝ)-p.1)*a i*monomial (k i) r)=
(∑ i, ((k i).1:ℝ)*a i*monomial (k i) r)-(∑ i,a i*monomial (k i) r)*p.1
rw [Finset.sum_mul,← Finset.sum_sub_distrib]
apply Finset.sum_congr rfl
intro i _
ring
· change (∑ i, (((k i).2:ℝ)-p.2)*a i*monomial (k i) r)=
(∑ i, ((k i).2:ℝ)*a i*monomial (k i) r)-(∑ i,a i*monomial (k i) r)*p.2
rw [Finset.sum_mul,← Finset.sum_sub_distrib]
apply Finset.sum_congr rfl
intro i _
ring
theorem centered_moment {k : ι → ℕ × ℕ} {a : ι → ℝ} {r : Plane}
(hP : polynomial k a r≠0) : centered k a (moment k a r) r=0 := by
rw [centered_eq,moment,smul_smul,mul_inv_cancel₀ hP,one_smul,sub_self]
def monomialDerivative (k : ℕ × ℕ) (r : Plane) : Plane →L[ℝ] ℝ :=
((k.1:ℝ)*r.1^(k.1-1)*r.2^k.2) • ContinuousLinearMap.fst ℝ ℝ ℝ +
(r.1^k.1*(k.2:ℝ)*r.2^(k.2-1)) • ContinuousLinearMap.snd ℝ ℝ ℝ
theorem monomial_hasFDerivAt (k : ℕ × ℕ) (r : Plane) :
HasFDerivAt (monomial k) (monomialDerivative k r) r := by
have h := (((ContinuousLinearMap.fst ℝ ℝ ℝ).hasFDerivAt (x:=r)).pow k.1).mul
(((ContinuousLinearMap.snd ℝ ℝ ℝ).hasFDerivAt (x:=r)).pow k.2)
convert! h using 1
apply ContinuousLinearMap.ext
intro v
simp [monomialDerivative]
ring
def centeredDerivative (k : ι → ℕ × ℕ) (a : ι → ℝ) (p r : Plane) : Plane →L[ℝ] Plane :=
(∑ i, ((((k i).1:ℝ)-p.1)*a i) • monomialDerivative (k i) r).prod
(∑ i, ((((k i).2:ℝ)-p.2)*a i) • monomialDerivative (k i) r)
theorem centered_hasFDerivAt (k : ι → ℕ × ℕ) (a : ι → ℝ) (p r : Plane) :
HasFDerivAt (centered k a p) (centeredDerivative k a p r) r := by
convert! (HasFDerivAt.sum (u := Finset.univ) (fun i _ => (monomial_hasFDerivAt (k i) r).const_mul
((((k i).1:ℝ)-p.1)*a i))).prodMk
(HasFDerivAt.sum (u := Finset.univ) (fun i _ => (monomial_hasFDerivAt (k i) r).const_mul
((((k i).2:ℝ)-p.2)*a i))) using 1
ext v <;> simp [centered]
theorem centeredDerivative_apply (k : ι → ℕ × ℕ) (a : ι → ℝ) (p r v : Plane) :
centeredDerivative k a p r v =
(∑ i, ((((k i).1:ℝ)-p.1)*a i)*
(((k i).1:ℝ)*r.1^((k i).1-1)*r.2^(k i).2*v.1+
r.1^(k i).1*((k i).2:ℝ)*r.2^((k i).2-1)*v.2),
∑ i, ((((k i).2:ℝ)-p.2)*a i)*
(((k i).1:ℝ)*r.1^((k i).1-1)*r.2^(k i).2*v.1+
r.1^(k i).1*((k i).2:ℝ)*r.2^((k i).2-1)*v.2)) := by
simp [centeredDerivative,monomialDerivative,mul_add,mul_assoc]
def faceMass (k : ℕ × ℕ) (a t : ℝ) : ℝ := if k.1=0 then a*t^k.2 else 0
def transverse (k : ι → ℕ × ℕ) (a : ι → ℝ) (t : ℝ) : ℝ :=
∑ i,if (k i).1=1 then a i*t^(k i).2 else 0
def active (k : ι → ℕ × ℕ) (a : ι → ℝ) (p t : ℝ) : ℝ :=
∑ i,if (k i).1=0 then (((k i).2:ℝ)-p)*a i*((k i).2:ℝ)*t^((k i).2-1) else 0
theorem zero_power_derivative (n : ℕ) :
(n:ℝ)*(0:ℝ)^(n-1)=if n=1 then 1 else 0 := by
rcases n with _|n
· simp
· rcases n with _|n
· simp
· simp
theorem monomial_axis (k : ℕ × ℕ) (t : ℝ) :
monomial k (0,t)=if k.1=0 then t^k.2 else 0 := by
by_cases h : k.1=0
· simp [monomial,h]
· simp [monomial,h,zero_pow h]
theorem monomialDerivative_axis (k : ℕ × ℕ) (t : ℝ) (v : Plane) :
monomialDerivative k (0,t) v=
(if k.1=1 then t^k.2 else 0)*v.1+
(if k.1=0 then (k.2:ℝ)*t^(k.2-1) else 0)*v.2 := by
simp only [monomialDerivative,add_apply,
smul_apply,
smul_eq_mul,zero_power_derivative]
by_cases h0 : k.1=0
· simp [h0]
· by_cases h1 : k.1=1
· simp [h1]
· simp [h0,h1,zero_pow h0]
theorem centered_axis_snd (k : ι → ℕ × ℕ) (a : ι → ℝ) (p : Plane) (t : ℝ) :
(centered k a p (0,t)).2=∑ i,(((k i).2:ℝ)-p.2)*faceMass (k i) (a i) t := by
apply Finset.sum_congr rfl
intro i _
rw [monomial_axis]
by_cases h : (k i).1=0 <;> simp [faceMass,h,mul_assoc]
theorem centeredDerivative_axis_fst (k : ι → ℕ × ℕ) (a : ι → ℝ) (p t : ℝ) (v : Plane) :
(centeredDerivative k a (0,p) (0,t) v).1=transverse k a t*v.1 := by
simp only [centeredDerivative,ContinuousLinearMap.prod_apply,
sum_apply,smul_apply,smul_eq_mul,
monomialDerivative_axis,sub_zero,transverse,Finset.sum_mul]
apply Finset.sum_congr rfl
intro i _
by_cases h0 : (k i).1=0
· simp [h0]
· by_cases h1 : (k i).1=1
· simp [h1,mul_assoc]
· simp [h0,h1]
theorem centeredDerivative_axis_snd (k : ι → ℕ × ℕ) (a : ι → ℝ) (p t u : ℝ) :
(centeredDerivative k a (0,p) (0,t) (0,u)).2=active k a p t*u := by
simp only [centeredDerivative,ContinuousLinearMap.prod_apply,
sum_apply,smul_apply,smul_eq_mul,
monomialDerivative_axis,mul_zero,zero_add,active,Finset.sum_mul]
apply Finset.sum_congr rfl
intro i _
by_cases h : (k i).1=0 <;> simp [h,mul_assoc]
theorem transverse_pos {k : ι → ℕ × ℕ} {a : ι → ℝ} (ha : ∀ i,0