Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Analysis.HardSphere.FormDomain.Basic
import OAI.Analysis.HardSphere.FormDomain.Energy

namespace OAI

noncomputable section
open MeasureTheory Filter Set
open scoped ENNReal Topology BigOperators
open scoped BigOperators ComplexConjugate
namespace HardSphere

theorem IsGroundVector.first_variation {N : ℕ} {L : Length} {a : ℝ}
    {u : Jet N L} (hu : IsGroundVector N L a u) (v : Jet N L)
    (hv : v ∈ dirichletDomain N L a) (hb : bosonic v.1) :
    (∑ p : Direction N, inner ℂ (u.2 p) (v.2 p)) =
      (energy u : ℂ) * inner ℂ u.1 v.1 := by
  apply Variational.ground_pairing (groundFormSubmodule N L a) u
  · exact mem_groundFormSubmodule.mpr ⟨hu.1, hu.2.1⟩
  · exact hu.2.2.1
  · intro w hw hn
    exact hu.2.2.2 w (mem_groundFormSubmodule.mp hw).1
      (mem_groundFormSubmodule.mp hw).2 hn
  · exact mem_groundFormSubmodule.mpr ⟨hv, hb⟩

theorem IsGroundVector.rayleigh {N : ℕ} {L : Length} {a : ℝ}
    {u : Jet N L} (hu : IsGroundVector N L a u) (v : Jet N L)
    (hv : v ∈ dirichletDomain N L a) (hb : bosonic v.1) :
    energy u * ‖v.1‖ ^ 2 ≤ energy v := by
  exact Variational.rayleigh (groundFormSubmodule N L a) u
    (fun w hw hn ↦ hu.2.2.2 w (mem_groundFormSubmodule.mp hw).1
      (mem_groundFormSubmodule.mp hw).2 hn) v
    (mem_groundFormSubmodule.mpr ⟨hv, hb⟩)

@[simp] lemma torusDistance_comm (L : Length) (x y : Torus L) :
    torusDistance L x y = torusDistance L y x := by
  simp only [torusDistance, dist_comm]

lemma allowed_iff_ne {N : ℕ} {L : Length} {a : ℝ} {X : Configuration N L} :
    X ∈ allowed N L a ↔ ∀ i j, i ≠ j → a < torusDistance L (X i) (X j) := by
  constructor
  · intro h i j hij
    rcases lt_or_gt_of_ne hij with hlt | hgt
    · exact h i j hlt
    · simpa only [torusDistance_comm] using h j i hgt
  · intro h i j hij
    exact h i j (ne_of_lt hij)

lemma allowed_permute {N : ℕ} {L : Length} {a : ℝ}
    (σ : Equiv.Perm (Fin N)) (X : Configuration N L) :
    permuteConfiguration σ X ∈ allowed N L a ↔ X ∈ allowed N L a := by
  simp only [allowed_iff_ne, permuteConfiguration]
  constructor
  · intro h i j hij
    simpa using h (σ.symm i) (σ.symm j) (σ.symm.injective.ne hij)
  · intro h i j hij
    exact h (σ i) (σ j) (σ.injective.ne hij)

def permuteLift {N : ℕ} (σ : Equiv.Perm (Fin N)) :
    LiftedConfiguration N ≃L[ℝ] LiftedConfiguration N where
  toFun X := fun i ↦ X (σ i)
  invFun X := fun i ↦ X (σ.symm i)
  left_inv X := by funext i; simp
  right_inv X := by funext i; simp
  map_add' _ _ := rfl
  map_smul' _ _ := rfl
  continuous_toFun := continuous_pi fun _ ↦ continuous_apply _
  continuous_invFun := continuous_pi fun _ ↦ continuous_apply _

lemma permuteLift_direction {N : ℕ} (σ : Equiv.Perm (Fin N)) (p : Direction N) :
    permuteLift σ (Pi.single p.1 (Pi.single p.2 1)) =
      Pi.single (σ.symm p.1) (Pi.single p.2 1) := by
  ext i j
  change (Pi.single p.1 (Pi.single p.2 (1 : ℝ)) : LiftedConfiguration N) (σ i) j = _
  by_cases h : i = σ.symm p.1
  · subst i; simp
  · have h' : σ i ≠ p.1 := fun hh ↦ h ((Equiv.eq_symm_apply σ).mpr hh)
    simp [Pi.single_eq_of_ne h, Pi.single_eq_of_ne h']

lemma coordDerivative_permute {N : ℕ} {L : Length} {a : ℝ}
    (f : SmoothTest N L a) (σ : Equiv.Perm (Fin N)) (p : Direction N) :
    coordDerivative N L (f.toFun ∘ permuteConfiguration σ) p =
      coordDerivative N L f.toFun (σ.symm p.1, p.2) ∘ permuteConfiguration σ := by
  funext X
  change (fderiv ℝ ((fun Y ↦ f.toFun (project N L Y)) ∘ permuteLift σ)
    (representative N L X)) (Pi.single p.1 (Pi.single p.2 1)) = _
  rw [fderiv_comp _ (f.smooth.differentiable (by simp)).differentiableAt
    (permuteLift σ).differentiableAt, (permuteLift σ).fderiv]
  simp only [ContinuousLinearMap.comp_apply, ContinuousLinearEquiv.coe_coe,
    permuteLift_direction]
  rfl

def SmoothTest.permute {N : ℕ} {L : Length} {a : ℝ}
    (f : SmoothTest N L a) (σ : Equiv.Perm (Fin N)) : SmoothTest N L a where
  toFun := f.toFun ∘ permuteConfiguration σ
  smooth := f.smooth.comp (permuteLift σ).contDiff
  compact := HasCompactSupport.of_compactSpace _
  support_allowed := by
    intro X hX
    apply (allowed_permute σ X).mp
    exact f.support_allowed
      (tsupport_comp_subset_preimage (f := permuteConfiguration σ) f.toFun (continuous_pi fun i ↦ continuous_apply (σ i)) hX)
  memLp := f.memLp.comp_measurePreserving (measurePreserving_permute σ)
  partial_memLp := by
    intro p
    rw [coordDerivative_permute]
    exact (f.partial_memLp (σ.symm p.1, p.2)).comp_measurePreserving
      (measurePreserving_permute σ)

section

attribute [local instance] scalarL2_addCommMonoid scalarL2_module

lemma continuous_permutationPullback_jet {N : ℕ} {L : Length} (σ : Equiv.Perm (Fin N)) :
    Continuous (fun u : Jet N L ↦ (permutationPullback σ u.1,
      fun direction : Direction N ↦
        permutationPullback σ (u.2 (σ.symm direction.1, direction.2)))) := by
  let pullback : ScalarL2 N L →ₗᵢ[ℂ] ScalarL2 N L :=
    Lp.compMeasurePreservingₗᵢ ℂ (permuteConfiguration σ) (measurePreserving_permute σ)
  exact (pullback.continuous.comp continuous_fst).prodMk
    (continuous_pi fun p ↦ pullback.continuous.comp ((continuous_apply _).comp continuous_snd))

def permutationJet {N : ℕ} {L : Length} (σ : Equiv.Perm (Fin N)) :
    Jet N L →L[ℂ] Jet N L where
  toFun u := (permutationPullback σ u.1,
    fun p ↦ permutationPullback σ (u.2 (σ.symm p.1, p.2)))
  map_add' _ _ := by simp [map_add]; rfl
  map_smul' _ _ := by simp [map_smul]; rfl
  cont := continuous_permutationPullback_jet σ

end

lemma SmoothTest.jet_permute {N : ℕ} {L : Length} {a : ℝ}
    (f : SmoothTest N L a) (σ : Equiv.Perm (Fin N)) :
    (f.permute σ).jet = permutationJet σ f.jet := by
  apply Prod.ext
  · apply Lp.ext
    exact (MemLp.coeFn_toLp (f.permute σ).memLp).trans
      ((Lp.coeFn_compMeasurePreserving f.jet.1 (measurePreserving_permute σ)).trans
        ((measurePreserving_permute σ).quasiMeasurePreserving.ae_eq_comp
          (MemLp.coeFn_toLp f.memLp))).symm
  · funext p
    apply Lp.ext
    have h := (measurePreserving_permute (L := L) σ).quasiMeasurePreserving.ae_eq_comp
      (MemLp.coeFn_toLp (f.partial_memLp (σ.symm p.1, p.2)))
    refine (MemLp.coeFn_toLp ((f.permute σ).partial_memLp p)).trans ?_
    change coordDerivative N L (f.toFun ∘ permuteConfiguration σ) p =ᵐ[volume] _
    rw [coordDerivative_permute]
    exact ((Lp.coeFn_compMeasurePreserving _ (measurePreserving_permute σ)).trans h).symm

lemma permutationJet_mem_domain {N : ℕ} {L : Length} {a : ℝ}
    (σ : Equiv.Perm (Fin N)) {u : Jet N L} (hu : u ∈ dirichletDomain N L a) :
    permutationJet σ u ∈ dirichletDomain N L a := by
  apply map_mem_closure (permutationJet σ).continuous hu
  rintro _ ⟨f, rfl⟩
  exact ⟨f.permute σ, f.jet_permute σ⟩

lemma energy_permutationJet {N : ℕ} {L : Length} (σ : Equiv.Perm (Fin N))
    (u : Jet N L) : energy (permutationJet σ u) = energy u := by
  change (∑ p : Direction N, ‖Lp.compMeasurePreserving _ (measurePreserving_permute σ)
    (u.2 (σ.symm p.1, p.2))‖ ^ 2) = _
  simp only [Lp.norm_compMeasurePreserving]
  exact Equiv.sum_comp (Equiv.prodCongr σ.symm (Equiv.refl (Fin 3)))
    (fun p ↦ ‖u.2 p‖ ^ 2)

end HardSphere

end

end OAI

	import Mathlib
	import OAI.Analysis.HardSphere.FormDomain.Basic
	import OAI.Analysis.HardSphere.FormDomain.Energy

	namespace OAI

	noncomputable section
	open MeasureTheory Filter Set
	open scoped ENNReal Topology BigOperators
	open scoped BigOperators ComplexConjugate
	namespace HardSphere

	theorem IsGroundVector.first_variation {N : ℕ} {L : Length} {a : ℝ}
	{u : Jet N L} (hu : IsGroundVector N L a u) (v : Jet N L)
	(hv : v ∈ dirichletDomain N L a) (hb : bosonic v.1) :
	(∑ p : Direction N, inner ℂ (u.2 p) (v.2 p)) =
	(energy u : ℂ) * inner ℂ u.1 v.1 := by
	apply Variational.ground_pairing (groundFormSubmodule N L a) u
	· exact mem_groundFormSubmodule.mpr ⟨hu.1, hu.2.1⟩
	· exact hu.2.2.1
	· intro w hw hn
	exact hu.2.2.2 w (mem_groundFormSubmodule.mp hw).1
	(mem_groundFormSubmodule.mp hw).2 hn
	· exact mem_groundFormSubmodule.mpr ⟨hv, hb⟩

	theorem IsGroundVector.rayleigh {N : ℕ} {L : Length} {a : ℝ}
	{u : Jet N L} (hu : IsGroundVector N L a u) (v : Jet N L)
	(hv : v ∈ dirichletDomain N L a) (hb : bosonic v.1) :
	energy u * ‖v.1‖ ^ 2 ≤ energy v := by
	exact Variational.rayleigh (groundFormSubmodule N L a) u
	(fun w hw hn ↦ hu.2.2.2 w (mem_groundFormSubmodule.mp hw).1
	(mem_groundFormSubmodule.mp hw).2 hn) v
	(mem_groundFormSubmodule.mpr ⟨hv, hb⟩)

	@[simp] lemma torusDistance_comm (L : Length) (x y : Torus L) :
	torusDistance L x y = torusDistance L y x := by
	simp only [torusDistance, dist_comm]

	lemma allowed_iff_ne {N : ℕ} {L : Length} {a : ℝ} {X : Configuration N L} :
	X ∈ allowed N L a ↔ ∀ i j, i ≠ j → a < torusDistance L (X i) (X j) := by
	constructor
	· intro h i j hij
	rcases lt_or_gt_of_ne hij with hlt \| hgt
	· exact h i j hlt
	· simpa only [torusDistance_comm] using h j i hgt
	· intro h i j hij
	exact h i j (ne_of_lt hij)

	lemma allowed_permute {N : ℕ} {L : Length} {a : ℝ}
	(σ : Equiv.Perm (Fin N)) (X : Configuration N L) :
	permuteConfiguration σ X ∈ allowed N L a ↔ X ∈ allowed N L a := by
	simp only [allowed_iff_ne, permuteConfiguration]
	constructor
	· intro h i j hij
	simpa using h (σ.symm i) (σ.symm j) (σ.symm.injective.ne hij)
	· intro h i j hij
	exact h (σ i) (σ j) (σ.injective.ne hij)

	def permuteLift {N : ℕ} (σ : Equiv.Perm (Fin N)) :
	LiftedConfiguration N ≃L[ℝ] LiftedConfiguration N where
	toFun X := fun i ↦ X (σ i)
	invFun X := fun i ↦ X (σ.symm i)
	left_inv X := by funext i; simp
	right_inv X := by funext i; simp
	map_add' _ _ := rfl
	map_smul' _ _ := rfl
	continuous_toFun := continuous_pi fun _ ↦ continuous_apply _
	continuous_invFun := continuous_pi fun _ ↦ continuous_apply _

	lemma permuteLift_direction {N : ℕ} (σ : Equiv.Perm (Fin N)) (p : Direction N) :
	permuteLift σ (Pi.single p.1 (Pi.single p.2 1)) =
	Pi.single (σ.symm p.1) (Pi.single p.2 1) := by
	ext i j
	change (Pi.single p.1 (Pi.single p.2 (1 : ℝ)) : LiftedConfiguration N) (σ i) j = _
	by_cases h : i = σ.symm p.1
	· subst i; simp
	· have h' : σ i ≠ p.1 := fun hh ↦ h ((Equiv.eq_symm_apply σ).mpr hh)
	simp [Pi.single_eq_of_ne h, Pi.single_eq_of_ne h']

	lemma coordDerivative_permute {N : ℕ} {L : Length} {a : ℝ}
	(f : SmoothTest N L a) (σ : Equiv.Perm (Fin N)) (p : Direction N) :
	coordDerivative N L (f.toFun ∘ permuteConfiguration σ) p =
	coordDerivative N L f.toFun (σ.symm p.1, p.2) ∘ permuteConfiguration σ := by
	funext X
	change (fderiv ℝ ((fun Y ↦ f.toFun (project N L Y)) ∘ permuteLift σ)
	(representative N L X)) (Pi.single p.1 (Pi.single p.2 1)) = _
	rw [fderiv_comp _ (f.smooth.differentiable (by simp)).differentiableAt
	(permuteLift σ).differentiableAt, (permuteLift σ).fderiv]
	simp only [ContinuousLinearMap.comp_apply, ContinuousLinearEquiv.coe_coe,
	permuteLift_direction]
	rfl

	def SmoothTest.permute {N : ℕ} {L : Length} {a : ℝ}
	(f : SmoothTest N L a) (σ : Equiv.Perm (Fin N)) : SmoothTest N L a where
	toFun := f.toFun ∘ permuteConfiguration σ
	smooth := f.smooth.comp (permuteLift σ).contDiff
	compact := HasCompactSupport.of_compactSpace _
	support_allowed := by
	intro X hX
	apply (allowed_permute σ X).mp
	exact f.support_allowed
	(tsupport_comp_subset_preimage (f := permuteConfiguration σ) f.toFun (continuous_pi fun i ↦ continuous_apply (σ i)) hX)
	memLp := f.memLp.comp_measurePreserving (measurePreserving_permute σ)
	partial_memLp := by
	intro p
	rw [coordDerivative_permute]
	exact (f.partial_memLp (σ.symm p.1, p.2)).comp_measurePreserving
	(measurePreserving_permute σ)

	section

	attribute [local instance] scalarL2_addCommMonoid scalarL2_module

	lemma continuous_permutationPullback_jet {N : ℕ} {L : Length} (σ : Equiv.Perm (Fin N)) :
	Continuous (fun u : Jet N L ↦ (permutationPullback σ u.1,
	fun direction : Direction N ↦
	permutationPullback σ (u.2 (σ.symm direction.1, direction.2)))) := by
	let pullback : ScalarL2 N L →ₗᵢ[ℂ] ScalarL2 N L :=
	Lp.compMeasurePreservingₗᵢ ℂ (permuteConfiguration σ) (measurePreserving_permute σ)
	exact (pullback.continuous.comp continuous_fst).prodMk
	(continuous_pi fun p ↦ pullback.continuous.comp ((continuous_apply _).comp continuous_snd))

	def permutationJet {N : ℕ} {L : Length} (σ : Equiv.Perm (Fin N)) :
	Jet N L →L[ℂ] Jet N L where
	toFun u := (permutationPullback σ u.1,
	fun p ↦ permutationPullback σ (u.2 (σ.symm p.1, p.2)))
	map_add' _ _ := by simp [map_add]; rfl
	map_smul' _ _ := by simp [map_smul]; rfl
	cont := continuous_permutationPullback_jet σ

	end

	lemma SmoothTest.jet_permute {N : ℕ} {L : Length} {a : ℝ}
	(f : SmoothTest N L a) (σ : Equiv.Perm (Fin N)) :
	(f.permute σ).jet = permutationJet σ f.jet := by
	apply Prod.ext
	· apply Lp.ext
	exact (MemLp.coeFn_toLp (f.permute σ).memLp).trans
	((Lp.coeFn_compMeasurePreserving f.jet.1 (measurePreserving_permute σ)).trans
	((measurePreserving_permute σ).quasiMeasurePreserving.ae_eq_comp
	(MemLp.coeFn_toLp f.memLp))).symm
	· funext p
	apply Lp.ext
	have h := (measurePreserving_permute (L := L) σ).quasiMeasurePreserving.ae_eq_comp
	(MemLp.coeFn_toLp (f.partial_memLp (σ.symm p.1, p.2)))
	refine (MemLp.coeFn_toLp ((f.permute σ).partial_memLp p)).trans ?_
	change coordDerivative N L (f.toFun ∘ permuteConfiguration σ) p =ᵐ[volume] _
	rw [coordDerivative_permute]
	exact ((Lp.coeFn_compMeasurePreserving _ (measurePreserving_permute σ)).trans h).symm

	lemma permutationJet_mem_domain {N : ℕ} {L : Length} {a : ℝ}
	(σ : Equiv.Perm (Fin N)) {u : Jet N L} (hu : u ∈ dirichletDomain N L a) :
	permutationJet σ u ∈ dirichletDomain N L a := by
	apply map_mem_closure (permutationJet σ).continuous hu
	rintro _ ⟨f, rfl⟩
	exact ⟨f.permute σ, f.jet_permute σ⟩

	lemma energy_permutationJet {N : ℕ} {L : Length} (σ : Equiv.Perm (Fin N))
	(u : Jet N L) : energy (permutationJet σ u) = energy u := by
	change (∑ p : Direction N, ‖Lp.compMeasurePreserving _ (measurePreserving_permute σ)
	(u.2 (σ.symm p.1, p.2))‖ ^ 2) = _
	simp only [Lp.norm_compMeasurePreserving]
	exact Equiv.sum_comp (Equiv.prodCongr σ.symm (Equiv.refl (Fin 3)))
	(fun p ↦ ‖u.2 p‖ ^ 2)

	end HardSphere

	end

	end OAI