import OAI.Geometry.Arnold.Homotopy

namespace OAI

universe uE uF

noncomputable section
open scoped BigOperators ContDiff
open MeasureTheory Set Filter
open scoped Matrix.Norms.Operator
open scoped Topology
open Function
open ContinuousAlternatingMap

namespace ArnoldCounterexample.CriticalBound
open ArnoldCounterexample.Forms ContinuousAlternatingMap

local instance : TopologicalSpace RM :=
  (inferInstance : NormedAddCommGroup RM).toMetricSpace.toPseudoMetricSpace.toUniformSpace.toTopologicalSpace

/-- The Haar integral of a closed horizontal cylinder form is independent of
its slice. The bracket condition is algebraic. -/
theorem haar_homotopy_invariant
    (Θ : ℝ × RM → (ℝ × RM) [⋀^Fin 6]→L[ℝ] ℝ)
    (hΘ : ContDiff ℝ ∞ Θ) (hclosed : ∀ p, extDeriv Θ p = 0)
    (hvertical : ∀ (t : ℝ) (A : O5) (i j : Fin 5),
      sliceContraction Θ t A
        (Matrix.vecCons ((A : RM) * (generators i.castSucc * generators j.succ -
          generators j.succ * generators i.castSucc))
          (j.removeNth (i.castSucc.removeNth (fun k ↦ (A : RM) * generators k)))) = 0)
    (a b : ℝ) :
    (∫ A : O5, Θ (a, A) (fun i ↦ (0, (A : RM) * generators i)) ∂haar) =
    (∫ A : O5, Θ (b, A) (fun i ↦ (0, (A : RM) * generators i)) ∂haar) := by
  let : CompactSpace O5 := o5_compactSpace
  let : BorelSpace O5 := o5_borelSpace
  let F : ℝ → O5 → ℝ := fun t A ↦ Θ (t, A) (fun i ↦ (0, (A : RM) * generators i))
  let D : ℝ → O5 → ℝ := fun t A ↦
    (fderiv ℝ Θ (t, A) (1, 0)) (fun i ↦ (0, (A : RM) * generators i))
  have hF : Continuous (Function.uncurry F) := by
    have h := contDiff_alternating_apply hΘ (v := fun i p ↦ (0, p.2 * generators i))
      (fun i ↦ (contDiff_const : ContDiff ℝ ∞ (fun _ : ℝ × RM ↦ (0 : ℝ))).prodMk
        (contDiff_snd.mul contDiff_const))
    exact h.continuous.comp (continuous_fst.prodMk (continuous_subtype_val.comp continuous_snd))
  have hD : Continuous (Function.uncurry D) := by
    have hd : ContDiff ℝ ∞ (fun p : ℝ × RM ↦ fderiv ℝ Θ p (1, 0)) :=
      (hΘ.fderiv_right (by simp)).clm_apply contDiff_const
    have h := contDiff_alternating_apply hd (v := fun i p ↦ (0, p.2 * generators i))
      (fun i ↦ (contDiff_const : ContDiff ℝ ∞ (fun _ : ℝ × RM ↦ (0 : ℝ))).prodMk
        (contDiff_snd.mul contDiff_const))
    exact h.continuous.comp (continuous_fst.prodMk (continuous_subtype_val.comp continuous_snd))
  have hd (t : ℝ) (A : O5) : HasDerivAt (fun s ↦ F s A) (D t A) t := by
    have h := ((ContinuousAlternatingMap.apply ℝ (ℝ × RM) ℝ
      (fun i ↦ (0, (A : RM) * generators i))).hasFDerivAt).comp_hasDerivAt t
        ((hΘ.differentiable (by simp) (t, A)).hasFDerivAt.comp_hasDerivAt t
          ((hasDerivAt_id t).prodMk (hasDerivAt_const t (A : RM))))
    exact h
  have dzero (t : ℝ) : ∫ A : O5, D t A ∂haar = 0 := by
    have hz := integral_extDeriv_rightMul_eq_zero (sliceContraction Θ t)
      (sliceContraction_contDiff Θ hΘ t) generators generators_skew (hvertical t)
    convert hz using 1
    apply integral_congr_ae
    exact Filter.Eventually.of_forall (fun A ↦
      (cylinder_homotopy_identity Θ (hΘ.differentiable (by simp)) t A
        (hclosed (t, A)) (fun i ↦ (A : RM) * generators i)).symm)
  have hder (t : ℝ) : HasDerivAt (fun s ↦ ∫ A, F s A ∂haar) 0 t := by
    convert hasDerivAt_integral_of_compact haar hF hD hd t using 1
    exact (dzero t).symm
  exact is_const_of_deriv_eq_zero (fun t ↦ (hder t).differentiableAt)
    (fun t ↦ (hder t).deriv) a b

end ArnoldCounterexample.CriticalBound

namespace ArnoldCounterexample.CriticalBound
open ArnoldCounterexample.Forms ContinuousAlternatingMap

lemma symplectic_velocities (i j : Fin 6) :
    symplectic 5 (velocities i) (velocities j) =
      if (i, j) ∈ ({(0, 1), (2, 3), (4, 5)} : Finset (Fin 6 × Fin 6)) then 1
      else if (j, i) ∈ ({(0, 1), (2, 3), (4, 5)} : Finset (Fin 6 × Fin 6)) then -1 else 0 := by
  fin_cases i <;> fin_cases j <;>
    simp [symplectic_apply, velocities, Fin.sum_univ_succ, Pi.single_apply,
      Complex.mul_re, Complex.mul_im, ← sq, baseScale_sq]

lemma cubeForm_velocities : cubeForm (symplectic 5) velocities = 48 := by
  simp only [cubeForm, pairForm_apply, squareForm_apply _ (symplectic_skew 5)]
  simp +decide [Fin.sum_univ_succ, Fin.removeNth, Fin.succAbove, symplectic_velocities]
  norm_num

end ArnoldCounterexample.CriticalBound

namespace ArnoldCounterexample.Forms
open ContinuousAlternatingMap

abbrev ProductSpace := CSpace 5 × CSpace 3

def productSymplectic : ProductSpace →L[ℝ] ProductSpace →L[ℝ] ℝ :=
  (symplectic 5).bilinearComp (.fst ℝ (CSpace 5) (CSpace 3)) (.fst ℝ (CSpace 5) (CSpace 3)) +
  (symplectic 3).bilinearComp (.snd ℝ (CSpace 5) (CSpace 3)) (.snd ℝ (CSpace 5) (CSpace 3))

@[simp] lemma productSymplectic_apply (u v : ProductSpace) :
    productSymplectic u v = symplectic 5 u.1 v.1 + symplectic 3 u.2 v.2 := rfl

lemma productSymplectic_skew (u v : ProductSpace) :
    productSymplectic u v = -productSymplectic v u := by
  simp only [productSymplectic_apply, symplectic_skew 5 u.1 v.1,
    symplectic_skew 3 u.2 v.2]
  ring

def productNormSq (z : ProductSpace) : ℝ := normSq 5 z.1 + normSq 3 z.2

lemma productNormSq_contDiff : ContDiff ℝ ∞ productNormSq :=
  (normSq_contDiff 5 |>.comp contDiff_fst).add (normSq_contDiff 3 |>.comp contDiff_snd)

lemma productNormSq_fderiv (z v : ProductSpace) :
    fderiv ℝ productNormSq z v = -productSymplectic (Complex.I • z) v := by
  have hn5 := (normSq_contDiff 5).differentiable (by simp)
  have hn3 := (normSq_contDiff 3).differentiable (by simp)
  have hd := ((hn5 z.1).hasFDerivAt.comp z hasFDerivAt_fst).add
    ((hn3 z.2).hasFDerivAt.comp z hasFDerivAt_snd)
  rw [show fderiv ℝ productNormSq z = _ from hd.fderiv]
  simp only [_root_.add_apply, ContinuousLinearMap.comp_apply, ContinuousLinearMap.coe_fst',
    ContinuousLinearMap.coe_snd', normSq_fderiv, productSymplectic_apply, Prod.smul_fst,
    Prod.smul_snd]
  ring

lemma productNormSq_curve_orthogonal {c : ℝ → ProductSpace} {v : ProductSpace} {t : ℝ}
    (hc : HasDerivAt c v t) (hunit : ∀ s, productNormSq (c s) = 1) :
    productSymplectic (Complex.I • c t) v = 0 := by
  have hd := (productNormSq_contDiff.differentiable (by simp) (c t)).hasFDerivAt.comp_hasDerivAt t hc
  have heq : (fun s ↦ productNormSq (c s)) = fun _ : ℝ ↦ 1 := funext hunit
  dsimp only [Function.comp_def] at hd
  rw [heq] at hd
  have hzero := hd.unique (hasDerivAt_const t 1)
  rw [productNormSq_fderiv] at hzero
  exact neg_eq_zero.mp hzero

lemma normSq_real_smul (n : ℕ) (a : ℝ) (z : CSpace n) :
    normSq n (a • z) = a ^ 2 * normSq n z := by
  simp [normSq, Pi.smul_apply, Complex.real_smul, Complex.normSq_mul,
    Complex.normSq_ofReal, Finset.mul_sum, sq]

/-- Differentiating honest circle equivariance, rather than assuming its
infinitesimal consequence. -/
lemma fderiv_circle_equivariant {m n : ℕ} (F : CSpace m → CSpace n)
    (hF : Differentiable ℝ F) (z : CSpace m)
    (hphase : ∀ a : ℂ, Complex.normSq a = 1 → F (a • z) = a • F z) :
    fderiv ℝ F z (Complex.I • z) = Complex.I • F z := by
  let a : ℝ → ℂ := fun t ↦ Complex.exp ((t : ℂ) * Complex.I)
  have ha : HasDerivAt a Complex.I 0 := by
    simpa [a] using ((Complex.ofRealCLM.hasDerivAt).mul_const Complex.I).cexp (x := (0 : ℝ))
  have ha0 : a 0 = 1 := by simp [a]
  have hau (t : ℝ) : Complex.normSq (a t) = 1 := by
    simp [a, Complex.normSq_eq_norm_sq, Complex.norm_exp]
  have heq : (fun t ↦ F (a t • z)) = fun t ↦ a t • F z :=
    funext (fun t ↦ hphase _ (hau t))
  have hd := (hF (a 0 • z)).hasFDerivAt.comp_hasDerivAt 0 (ha.smul_const z)
  rw [ha0, one_smul] at hd
  dsimp only [Function.comp_def] at hd
  rw [heq] at hd
  exact hd.unique (ha.smul_const (F z))

end ArnoldCounterexample.Forms

namespace ArnoldCounterexample.Forms

lemma time_slice_hasDerivAt {E : Type uE} {F : Type uF} [NormedAddCommGroup E] [NormedSpace ℝ E]
    [NormedAddCommGroup F] [NormedSpace ℝ F]
    (H : ℝ × E → F) (hH : Differentiable ℝ H) (x : E) (t : ℝ) :
    HasDerivAt (fun s ↦ H (s, x)) (fderiv ℝ H (t, x) (1, 0)) t := by
  exact (hH (t, x)).hasFDerivAt.comp_hasDerivAt t
    ((hasDerivAt_id t).prodMk (hasDerivAt_const t x))

lemma productNormSq_time_tangent {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (H : ℝ × E → ProductSpace) (hH : ContDiff ℝ ∞ H) (x : E)
    (hunit : ∀ s, productNormSq (H (s, x)) = 1) (t : ℝ) :
    productSymplectic (Complex.I • H (t, x)) (fderiv ℝ H (t, x) (1, 0)) = 0 := by
  apply productNormSq_curve_orthogonal
    (time_slice_hasDerivAt H (hH.differentiable (by simp)) x t) hunit

end ArnoldCounterexample.Forms

namespace ArnoldCounterexample.Forms
open ContinuousAlternatingMap

lemma alternating_eq_zero_on_kernel
    {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
    {n : ℕ} (hdim : Module.finrank ℝ E ≤ n)
    (θ : E [⋀^Fin n]→L[ℝ] ℝ) (L : E →L[ℝ] ℝ) (hL : L ≠ 0)
    (v : Fin n → E) (hv : ∀ i, L (v i) = 0) : θ v = 0 := by
  apply θ.toAlternatingMap.map_linearDependent
  intro hLI
  let v' : Fin n → L.ker := fun i ↦ ⟨v i, hv i⟩
  have hLI' : LinearIndependent ℝ v' := hLI.of_comp L.ker.subtype
  have htop : L.ker ≠ ⊤ := by
    intro ht
    apply hL
    ext x
    have hx : x ∈ L.ker := by rw [ht]; trivial
    exact hx
  have hc := hLI'.fintype_card_le_finrank
  have hlt := Submodule.finrank_lt htop
  simp only [Fintype.card_fin] at hc
  omega

/-- Six tangent vectors of the unit five-sphere are necessarily dependent. -/
lemma six_form_vanishes_on_unit_sphere
    (θ : CSpace 3 [⋀^Fin 6]→L[ℝ] ℝ) (z : CSpace 3) (hz : normSq 3 z = 1)
    (v : Fin 6 → CSpace 3) (hv : ∀ i, fderiv ℝ (normSq 3) z (v i) = 0) :
    θ v = 0 := by
  apply alternating_eq_zero_on_kernel (L := fderiv ℝ (normSq 3) z) ?_ θ ?_ v hv
  · simp [CSpace, Module.finrank_pi_fintype, Complex.finrank_real_complex]
  · intro h
    have hs := normSq_fderiv 3 z z
    rw [h] at hs
    simp only [_root_.zero_apply, symplectic_I_left] at hs
    have heq : (∑ i : Fin 3, ((z i).re * (z i).re + (z i).im * (z i).im)) = 1 := by
      simpa only [normSq, Complex.normSq_apply] using hz
    rw [heq] at hs
    norm_num at hs

end ArnoldCounterexample.Forms

namespace ArnoldCounterexample.Forms
open ContinuousAlternatingMap

lemma cubeForm_second_radical_on {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (B : E →L[ℝ] E →L[ℝ] ℝ) (hskew : ∀ u v, B u v = -B v u)
    (u w : E) (v : Fin 4 → E) (hw : B u w = 0) (hv : ∀ i, B u (v i) = 0) :
    cubeForm B (Matrix.vecCons w (Matrix.vecCons u v)) = 0 := by
  have hz := cubeForm_radical_on B hskew u (Matrix.vecCons w v) (by
    intro i
    refine Fin.cases hw (fun j ↦ hv j) i)
  have heq : (Matrix.vecCons w (Matrix.vecCons u v)) ∘ Equiv.swap (0 : Fin 6) 1 =
      Matrix.vecCons u (Matrix.vecCons w v) := by
    ext i
    fin_cases i <;> simp [Equiv.swap_apply_def]
  have hs := (cubeForm B).toAlternatingMap.map_swap
    (Matrix.vecCons w (Matrix.vecCons u v)) (show (0 : Fin 6) ≠ 1 by decide)
  rw [heq] at hs
  change cubeForm B (Matrix.vecCons u (Matrix.vecCons w v)) =
    -cubeForm B (Matrix.vecCons w (Matrix.vecCons u v)) at hs
  rw [hz] at hs
  exact neg_eq_zero.mp hs.symm

end ArnoldCounterexample.Forms

end

end OAI