import Mathlib

namespace OAI

noncomputable section
namespace PoissonSupport.DeGiorgi
open MeasureTheory
open scoped ENNReal

abbrev Space (n : ℕ) := EuclideanSpace ℝ (Fin n)

def potential (t : ℝ) : ℝ := (1 - t ^ 2) ^ 2 / 4

def profile (s : ℝ) : ℝ := Real.tanh (s / Real.sqrt 2)

def surfaceTension : ℝ := 2 * Real.sqrt 2 / 3

def AllenCahn {n : ℕ} (v : Space n → ℝ) : Prop :=
  ∀ x, Laplacian.laplacian v x = v x ^ 3 - v x

def SecondVariation {n : ℕ} (v φ : Space n → ℝ) : ℝ :=
  ∫ x, (‖gradient φ x‖ ^ 2 + (3 * v x ^ 2 - 1) * φ x ^ 2)

def Stable {n : ℕ} (v : Space n → ℝ) : Prop :=
  ∀ φ : Space n → ℝ, ContDiff ℝ 1 φ → HasCompactSupport φ →
    0 ≤ SecondVariation v φ

def Energy {n : ℕ} (v : Space n → ℝ) (Ω : Set (Space n)) : ℝ :=
  ∫ x in Ω, (‖gradient v x‖ ^ 2 / 2 + potential (v x))

def unitBallVolume (m : ℕ) : ℝ :=
  volume.real (Metric.ball (0 : Space m) 1)

def density {n : ℕ} (v : Space n → ℝ) (R : ℝ) (y : Space n) : ℝ :=
  Energy v (Metric.ball y R) / (surfaceTension * unitBallVolume (n - 1) * R ^ (n - 1))

def densityAtInfinity {n : ℕ} (v : Space n → ℝ) : ℝ≥0∞ :=
  ⨆ R : { R : ℝ // 0 < R }, ENNReal.ofReal (density v R.val 0)

def Classified {n : ℕ} (v : Space n → ℝ) : Prop :=
  (∀ x, v x = 1) ∨ (∀ x, v x = -1) ∨
    ∃ (a : Space n) (b : ℝ), ‖a‖ = 1 ∧ ∀ x, v x = profile (inner ℝ a x - b)

def StableRigiditySeven : Prop :=
  ∀ v : Space 7 → ℝ, ContDiff ℝ 2 v →
    (∀ x, v x ∈ Set.Icc (-1 : ℝ) 1) → AllenCahn v → Stable v → Classified v

def MonotoneRigidityEight : Prop :=
  ∀ u : Space 8 → ℝ, ContDiff ℝ 2 u →
    (∀ x, u x ∈ Set.Ioo (-1 : ℝ) 1) → AllenCahn u →
    (∀ x, 0 < fderiv ℝ u x (EuclideanSpace.single (7 : Fin 8) 1)) →
    ∃ (e : Space 8) (c : ℝ), ‖e‖ = 1 ∧ 0 < e 7 ∧
      ∀ x, u x = profile (inner ℝ e x - c)

def FiniteDensityRigidity : Prop :=
  ∀ (n : ℕ), 4 ≤ n → n ≤ 7 →
    ∀ v : Space n → ℝ, ContDiff ℝ 2 v →
      (∀ x, v x ∈ Set.Icc (-1 : ℝ) 1) → AllenCahn v → Stable v →
      densityAtInfinity v < ⊤ → Classified v

end PoissonSupport.DeGiorgi
end

noncomputable section
namespace PoissonSupport.DeGiorgi
open MeasureTheory
open scoped ENNReal
open Filter Topology

lemma second_deriv_nonneg_of_local_min {f : ℝ → ℝ} {x : ℝ}
    (hf : ContinuousAt f x) (hm : IsLocalMin f x) : 0 ≤ deriv (deriv f) x := by
  by_contra h
  have hn : deriv (deriv f) x < 0 := lt_of_not_ge h
  have hmax := isLocalMax_of_deriv_deriv_neg hn hm.deriv_eq_zero hf
  have he : f =ᶠ[𝓝 x] fun _ => f x := by
    filter_upwards [hm, hmax] with y hy hz using le_antisymm hz hy
  have hz := he.deriv.deriv_eq
  have hz0 : deriv (deriv f) x = 0 := by
    calc deriv (deriv f) x = deriv (deriv (fun _ : ℝ => f x)) x := hz
         _ = 0 := by simp
  linarith

def basis (n : ℕ) := EuclideanSpace.basisFun (Fin n) ℝ

def coordDeriv {n : ℕ} (f : Space n → ℝ) (i : Fin n) (x : Space n) : ℝ :=
  fderiv ℝ f x (basis n i)

lemma coordDeriv_contDiff {n : ℕ} {f : Space n → ℝ}
    (hf : ContDiff ℝ 2 f) (i : Fin n) : ContDiff ℝ 1 (coordDeriv f i) := by
  exact (hf.fderiv_right (m := 1) (by norm_num)).clm_apply contDiff_const

lemma laplacian_eq_sum_coordDeriv {n : ℕ} {f : Space n → ℝ} (hf : ContDiff ℝ 2 f)
    (x : Space n) :
    Laplacian.laplacian f x = ∑ i : Fin n, coordDeriv (coordDeriv f i) i x := by
  rw [InnerProductSpace.laplacian_eq_iteratedFDeriv_orthonormalBasis f (basis n)]
  apply Finset.sum_congr rfl
  intro i hi
  rw [iteratedFDeriv_two_apply]
  change _ = fderiv ℝ (fun y => fderiv ℝ f y (basis n i)) x (basis n i)
  simp [fderiv_clm_apply
    ((hf.fderiv_right (m := 1) (by norm_num)).differentiable (by norm_num) x)
    (differentiableAt_const (c := basis n i))]

lemma deriv_line {n : ℕ} {f : Space n → ℝ} (hf : Differentiable ℝ f)
    (x e : Space n) (t : ℝ) :
    deriv (fun s : ℝ => f (x + s • e)) t = fderiv ℝ f (x + t • e) e := by
  have hd : HasDerivAt (fun s : ℝ => x + s • e) e t := by
    simpa using (hasDerivAt_const t x).fun_add ((hasDerivAt_id t).smul_const e)
  exact ((hf _).hasFDerivAt.comp_hasDerivAt t hd).deriv

lemma laplacian_nonneg_of_local_min {n : ℕ} {f : Space n → ℝ}
    (hf : ContDiff ℝ 2 f) {x : Space n} (hm : IsLocalMin f x) :
    0 ≤ Laplacian.laplacian f x := by
  rw [laplacian_eq_sum_coordDeriv hf]
  apply Finset.sum_nonneg
  intro i _
  let g : ℝ → ℝ := fun s => f (x + s • basis n i)
  have hg : ContinuousAt g 0 := hf.continuous.continuousAt.comp
    (continuous_const.add (continuous_id.smul continuous_const)).continuousAt
  have hmin : IsLocalMin g 0 := by
    have ht : Tendsto (fun s : ℝ => x + s • basis n i) (𝓝 0) (𝓝 x) := by
      have hc : Continuous (fun s : ℝ => x + s • basis n i) :=
        continuous_const.add (continuous_id.smul continuous_const)
      simpa only [ContinuousAt, zero_smul, add_zero] using hc.continuousAt (x := 0)
    change ∀ᶠ s in 𝓝 (0 : ℝ), g 0 ≤ g s
    simpa [g] using ht.eventually hm
  have hder : deriv g = fun s => coordDeriv f i (x + s • basis n i) := by
    funext s
    exact deriv_line (hf.differentiable (by norm_num)) x (basis n i) s
  have hh := second_deriv_nonneg_of_local_min hg hmin
  rw [hder, deriv_line ((coordDeriv_contDiff hf i).differentiable (by norm_num))] at hh
  simpa [coordDeriv] using hh

lemma laplacian_nonpos_of_local_max {n : ℕ} {f : Space n → ℝ}
    (hf : ContDiff ℝ 2 f) {x : Space n} (hm : IsLocalMax f x) :
    Laplacian.laplacian f x ≤ 0 := by
  have h := laplacian_nonneg_of_local_min hf.neg hm.neg
  change 0 ≤ Laplacian.laplacian (-f) x at h
  rw [InnerProductSpace.laplacian_neg] at h
  simpa using h

lemma coordDeriv_translate {n : ℕ} {f : Space n → ℝ}
    (hf : Differentiable ℝ f) (a : Space n) (i : Fin n) (x : Space n) :
    coordDeriv (fun z => f (a + z)) i x = coordDeriv f i (a + x) := by
  dsimp [coordDeriv]
  have hd : HasFDerivAt (fun z => f (a + z)) (fderiv ℝ f (a + x)) x := by
    simpa only [Function.comp_def, zero_add, ContinuousLinearMap.comp_id, id_eq] using (hf (a + x)).hasFDerivAt.comp x
      ((hasFDerivAt_const a x).fun_add (hasFDerivAt_id x))
  rw [hd.fderiv]

lemma laplacian_translate {n : ℕ} {f : Space n → ℝ}
    (hf : ContDiff ℝ 2 f) (a x : Space n) :
    Laplacian.laplacian (fun z => f (a + z)) x = Laplacian.laplacian f (a + x) := by
  have hs : ContDiff ℝ 2 (fun z => f (a + z)) :=
    hf.comp (contDiff_const.add contDiff_id)
  rw [laplacian_eq_sum_coordDeriv hs,
    laplacian_eq_sum_coordDeriv hf]
  apply Finset.sum_congr rfl
  intro i _
  have hfun : coordDeriv (fun z => f (a + z)) i = fun z => coordDeriv f i (a + z) :=
    funext (coordDeriv_translate (hf.differentiable (by norm_num)) a i)
  rw [hfun, coordDeriv_translate ((coordDeriv_contDiff hf i).differentiable (by norm_num))]

lemma laplacian_norm_sq {n : ℕ} (x : Space n) :
    Laplacian.laplacian (fun z : Space n => ‖z‖ ^ 2) x = 2 * n := by
  have hs : ContDiff ℝ 2 (fun z : Space n => ‖z‖ ^ 2) := contDiff_id.norm_sq ℝ
  rw [laplacian_eq_sum_coordDeriv hs]
  have hfirst (i : Fin n) : coordDeriv (fun z : Space n => ‖z‖ ^ 2) i =
      fun z => 2 * inner ℝ z (basis n i) := by
    funext z
    simp [coordDeriv]
  simp_rw [hfirst]
  have hsecond (i : Fin n) : coordDeriv (fun z => 2 * inner ℝ z (basis n i)) i x = 2 := by
    dsimp [coordDeriv]
    have hd : HasFDerivAt (fun z : Space n => 2 * inner ℝ z (basis n i))
        ((2 : ℝ) • innerSL ℝ (basis n i)) x := by
      simpa [real_inner_comm] using
        ((innerSL ℝ (basis n i)).hasFDerivAt (x := x)).const_mul 2
    rw [hd.fderiv]
    simp [basis]
  simp_rw [hsecond]
  simp [mul_comm]

lemma hasFDerivAt_norm_nonzero {n : ℕ} {x : Space n} (hx : x ≠ 0) :
    HasFDerivAt (fun z : Space n => ‖z‖) (‖x‖⁻¹ • innerSL ℝ x) x := by
  have h := (hasStrictFDerivAt_norm_sq x).hasFDerivAt.sqrt
    (pow_ne_zero 2 (norm_ne_zero_iff.mpr hx))
  simp only [Real.sqrt_sq (norm_nonneg _)] at h
  have he : (1 / (2 * ‖x‖) : ℝ) * 2 = ‖x‖⁻¹ := by field_simp
  rw [← Nat.cast_smul_eq_nsmul ℝ 2, smul_smul, Nat.cast_ofNat, he] at h
  exact h

lemma laplacian_fun_sub {n : ℕ} {f g : Space n → ℝ}
    (hf : ContDiff ℝ 2 f) (hg : ContDiff ℝ 2 g) (x : Space n) :
    Laplacian.laplacian (fun z => f z - g z) x =
      Laplacian.laplacian f x - Laplacian.laplacian g x := by
  change Laplacian.laplacian (f - g) x = _
  exact hf.contDiffAt.laplacian_sub hg.contDiffAt

lemma laplacian_fun_add {n : ℕ} {f g : Space n → ℝ}
    (hf : ContDiff ℝ 2 f) (hg : ContDiff ℝ 2 g) (x : Space n) :
    Laplacian.laplacian (fun z => f z + g z) x =
      Laplacian.laplacian f x + Laplacian.laplacian g x := by
  change Laplacian.laplacian (f + g) x = _
  exact hf.contDiffAt.laplacian_add hg.contDiffAt

lemma laplacian_fun_const_mul {n : ℕ} {f : Space n → ℝ}
    (hf : ContDiff ℝ 2 f) (c : ℝ) (x : Space n) :
    Laplacian.laplacian (fun z => c * f z) x = c * Laplacian.laplacian f x := by
  change Laplacian.laplacian (c • f) x = _
  exact InnerProductSpace.laplacian_smul c hf.contDiffAt

lemma laplacian_comp_isometry {n : ℕ} {f : Space n → ℝ}
    (hf : ContDiff ℝ 2 f) (J : Space n ≃ₗᵢ[ℝ] Space n) (x : Space n) :
    Laplacian.laplacian (fun z => f (J z)) x = Laplacian.laplacian f (J x) := by
  rw [InnerProductSpace.laplacian_eq_iteratedFDeriv_orthonormalBasis _ (basis n),
    InnerProductSpace.laplacian_eq_iteratedFDeriv_orthonormalBasis _ ((basis n).map J)]
  apply Finset.sum_congr rfl
  intro i _
  change iteratedFDeriv ℝ 2 (f ∘ J.toContinuousLinearMap) x _ = _
  rw [J.toContinuousLinearMap.iteratedFDeriv_comp_right hf x (by norm_num)]
  simp only [ContinuousMultilinearMap.compContinuousLinearMap_apply,
    OrthonormalBasis.map_apply]
  congr 1
  ext j
  fin_cases j <;> simp

def reflect {n : ℕ} (e : Space n) : Space n ≃ₗᵢ[ℝ] Space n :=
  (Submodule.span ℝ {e})ᗮ.reflection

lemma reflect_apply {n : ℕ} {e : Space n} (he : ‖e‖ = 1) (z : Space n) :
    reflect e z = z - (2 * inner ℝ e z) • e := by
  simp only [reflect, Submodule.reflection_orthogonal_apply,
    Submodule.reflection_singleton_apply, he]
  norm_num only [RCLike.ofReal_real_eq_id, id_eq, one_pow, div_one] 
  rw [neg_sub, ← smul_assoc]
  simp only [two_smul, two_mul]

def poissonPenalty (A t : ℝ) : ℝ := A * (1 - (max (1 - t) 0) ^ 2)

lemma poissonPenalty_of_le (A : ℝ) {t : ℝ} (ht : t ≤ 1) :
    poissonPenalty A t = A * (2 * t - t ^ 2) := by
  rw [poissonPenalty, max_eq_left (by linarith)]
  ring

lemma poissonPenalty_of_ge (A : ℝ) {t : ℝ} (ht : 1 ≤ t) :
    poissonPenalty A t = A := by
  simp [poissonPenalty, max_eq_right (show 1 - t ≤ 0 by linarith)]

lemma poissonPenalty_nonneg {A t : ℝ} (hA : 0 ≤ A) (ht : 0 ≤ t) :
    0 ≤ poissonPenalty A t := by
  by_cases h : t ≤ 1
  · rw [poissonPenalty_of_le A h]
    apply mul_nonneg hA
    nlinarith
  · rw [poissonPenalty_of_ge A (le_of_not_ge h)]
    exact hA

lemma poissonPenalty_le_linear {A t : ℝ} (hA : 0 ≤ A) (ht : 0 ≤ t) :
    poissonPenalty A t ≤ 2 * A * t := by
  by_cases h : t ≤ 1
  · rw [poissonPenalty_of_le A h]
    nlinarith [mul_nonneg hA (sq_nonneg t)]
  · rw [poissonPenalty_of_ge A (le_of_not_ge h)]
    nlinarith

lemma poisson_maximum {n : ℕ} {f : Space n → ℝ} (hf : Continuous f)
    {M A ε : ℝ} (hM : 0 ≤ M) (hbound : ∀ x, |f x| ≤ M)
    (hA : 0 ≤ A) (hε : 0 < ε) :
    ∃ x y : Space n, ∀ a b : Space n,
      f b - f a - poissonPenalty A ‖b - a‖ - ε * (‖a‖ ^ 2 + ‖b‖ ^ 2) ≤
      f y - f x - poissonPenalty A ‖y - x‖ - ε * (‖x‖ ^ 2 + ‖y‖ ^ 2) := by
  let H : Space n × Space n → ℝ := fun z =>
    f z.2 - f z.1 - poissonPenalty A ‖z.2 - z.1‖ - ε * (‖z.1‖ ^ 2 + ‖z.2‖ ^ 2)
  have hc : Continuous H := by dsimp [H, poissonPenalty]; fun_prop
  let R := Real.sqrt (2 * M / ε + 1)
  have hR : 0 ≤ R := Real.sqrt_nonneg _
  have hR2 : R ^ 2 = 2 * M / ε + 1 := Real.sq_sqrt (by positivity)
  have hevent : ∀ᶠ z in cocompact (Space n × Space n), H z ≤ H (0, 0) := by
    filter_upwards [((isCompact_closedBall (0 : Space n) R).prod
      (isCompact_closedBall (0 : Space n) R)).compl_mem_cocompact] with z hz
    have hz' : R < ‖z.1‖ ∨ R < ‖z.2‖ := by
      simpa only [Set.mem_compl_iff, Set.mem_prod, Metric.mem_closedBall, dist_zero_right,
        not_and_or, not_le] using hz
    have hsq : R ^ 2 < ‖z.1‖ ^ 2 + ‖z.2‖ ^ 2 := by
      rcases hz' with hz' | hz'
      · nlinarith [sq_nonneg ‖z.2‖, norm_nonneg z.1]
      · nlinarith [sq_nonneg ‖z.1‖, norm_nonneg z.2]
    have heq : ε * (2 * M / ε + 1) = 2 * M + ε := by field_simp
    have hh := mul_lt_mul_of_pos_left hsq hε
    have hb1 := (abs_le.mp (hbound z.1)).1
    have hb2 := (abs_le.mp (hbound z.2)).2
    have hcost := poissonPenalty_nonneg hA (norm_nonneg (z.2 - z.1))
    dsimp [H]
    simp only [sub_self, norm_zero, poissonPenalty, sub_zero, max_eq_left zero_le_one,
      one_pow, mul_zero, zero_pow two_ne_zero, add_zero]
    rw [hR2, heq] at hh
    dsimp [poissonPenalty] at hcost
    linarith
  obtain ⟨⟨x, y⟩, hmax⟩ := hc.exists_forall_ge' (0, 0) hevent
  exact ⟨x, y, fun a b => hmax (a, b)⟩

lemma coordDeriv_affine {n : ℕ} (a : Space n) (c : ℝ) (i : Fin n) (x : Space n) :
    coordDeriv (fun z => c + inner ℝ a z) i x = inner ℝ a (basis n i) := by
  have hd : HasFDerivAt (fun z => c + inner ℝ a z) (innerSL ℝ a) x :=
    (innerSL ℝ a).hasFDerivAt.const_add c
  simp [coordDeriv, hd.fderiv]

lemma laplacian_affine {n : ℕ} (a : Space n) (c : ℝ) (x : Space n) :
    Laplacian.laplacian (fun z => c + inner ℝ a z) x = 0 := by
  have hc : ContDiff ℝ 2 (fun z : Space n => c + inner ℝ a z)  := contDiff_const.add ((innerSL ℝ a).contDiff)
  rw [laplacian_eq_sum_coordDeriv hc]
  have he (i : Fin n) : coordDeriv (fun z : Space n => c + inner ℝ a z) i =
      fun _ => inner ℝ a (basis n i) := funext (coordDeriv_affine a c i)
  simp_rw [he]
  simp [coordDeriv]

lemma laplacian_affine_sq {n : ℕ} (a : Space n) (c : ℝ) (x : Space n) :
    Laplacian.laplacian (fun z => (c + inner ℝ a z) ^ 2) x = 2 * ‖a‖ ^ 2 := by
  have hc : ContDiff ℝ 2 (fun z : Space n => (c + inner ℝ a z) ^ 2) :=
    (contDiff_const.add ((innerSL ℝ a).contDiff)).pow 2
  rw [laplacian_eq_sum_coordDeriv hc]
  have hfirst (i : Fin n) : coordDeriv (fun z : Space n => (c + inner ℝ a z) ^ 2) i =
      fun z => 2 * (c + inner ℝ a z) * inner ℝ a (basis n i) := by
    funext z
    have hd := ((innerSL ℝ a).hasFDerivAt (x := z)).const_add c
    have hd2 := hd.pow 2
    dsimp only [coordDeriv]
    change (fderiv ℝ (fun z => (c + (innerSL ℝ a) z) ^ 2) z) _ = _
    rw [hd2.fderiv]
    simp only [smul_apply, smul_eq_mul, Nat.cast_ofNat, Nat.reduceSub, pow_one, innerSL_apply_apply, nsmul_eq_mul]
  simp_rw [hfirst]
  have hsecond (i : Fin n) : coordDeriv
      (fun z => 2 * (c + inner ℝ a z) * inner ℝ a (basis n i)) i x =
      2 * (inner ℝ a (basis n i)) ^ 2 := by
    have hd := ((((innerSL ℝ a).hasFDerivAt (x := x)).const_add c).const_mul 2).mul_const
      (inner ℝ a (basis n i))
    dsimp only [coordDeriv]
    change (fderiv ℝ (fun z => 2 * (c + (innerSL ℝ a) z) * inner ℝ a ((basis n) i)) x) _ = _
    rw [hd.fderiv]
    simp [pow_two, mul_left_comm]
  simp_rw [hsecond]
  rw [← Finset.mul_sum]
  congr 1
  have hb := (basis n).sum_inner_mul_inner a a
  simpa only [real_inner_comm, ← pow_two, real_inner_self_eq_norm_sq] using hb

end PoissonSupport.DeGiorgi
end

end OAI