import OAI.Geometry.Arnold.Obstruction namespace OAI noncomputable section open scoped BigOperators ContDiff open scoped BigOperators ContDiff namespace ArnoldCounterexample namespace ThreePoints abbrev R3 := Fin 3 → ℝ abbrev Pair := R3 × R3 def sqNorm (x : R3) : ℝ := x 0 ^ 2 + x 1 ^ 2 + x 2 ^ 2 def sphereProduct : Set Pair := {q | sqNorm q.1 = 1 ∧ sqNorm q.2 = 1} def pole : R3 := ![0, 0, 1] def east : R3 := ![1, 0, 0] /-- A polynomial with three constrained critical points. -/ def f (q : Pair) : ℝ := q.1 0 + q.1 0 * q.2 0 + q.1 1 * q.2 1 + (q.1 2 - 1) * q.2 2 /-- The constrained critical equations of the polynomial. -/ def MultiplierCritical (q : Pair) : Prop := q ∈ sphereProduct ∧ ∃ lam μ : ℝ, 1 + q.2 0 = lam * q.1 0 ∧ q.2 1 = lam * q.1 1 ∧ q.2 2 = lam * q.1 2 ∧ q.1 0 = μ * q.2 0 ∧ q.1 1 = μ * q.2 1 ∧ q.1 2 - 1 = μ * q.2 2 def qzero : Pair := (pole, -east) def qplus : Pair := (![Real.sqrt 3 / 2, 0, -1/2], ![1/2, 0, -Real.sqrt 3 / 2]) def qminus : Pair := (![-Real.sqrt 3 / 2, 0, -1/2], ![1/2, 0, Real.sqrt 3 / 2]) /-- Algebraic elimination of every solution, without a nondegeneracy assumption. We eliminate using μ instead of t=|x-p|; μ²=t², so the remaining factor is μ²=3. -/ lemma classify_coordinates {x₀ x₁ x₂ y₀ y₁ y₂ lam μ : ℝ} (hx : x₀^2 + x₁^2 + x₂^2 = 1) (hy : y₀^2 + y₁^2 + y₂^2 = 1) (h₀ : 1 + y₀ = lam * x₀) (h₁ : y₁ = lam * x₁) (h₂ : y₂ = lam * x₂) (g₀ : x₀ = μ * y₀) (g₁ : x₁ = μ * y₁) (g₂ : x₂ - 1 = μ * y₂) : (x₀ = 0 ∧ x₁ = 0 ∧ x₂ = 1 ∧ y₀ = -1 ∧ y₁ = 0 ∧ y₂ = 0) ∨ (μ^2 = 3 ∧ x₀ = μ/2 ∧ x₁ = 0 ∧ x₂ = -1/2 ∧ y₀ = 1/2 ∧ y₁ = 0 ∧ y₂ = -μ/2) := by by_cases hμ : μ = 0 · left have hx₀ : x₀ = 0 := by simpa [hμ] using g₀ have hx₁ : x₁ = 0 := by simpa [hμ] using g₁ have hx₂ : x₂ = 1 := by simp [hμ] at g₂; linarith have hy₀ : y₀ = -1 := by rw [hx₀] at h₀; linarith have hy₁ : y₁ = 0 := by nlinarith [sq_nonneg y₂] have hy₂ : y₂ = 0 := by nlinarith [sq_nonneg y₁] exact ⟨hx₀, hx₁, hx₂, hy₀, hy₁, hy₂⟩ · right have gx₂ : x₂ = 1 + μ * y₂ := by linarith have hn : μ * (μ + 2 * y₂) = 0 := by have he : μ^2 * (y₀^2 + y₁^2 + y₂^2 - 1) + μ * (μ + 2 * y₂) = x₀^2 + x₁^2 + x₂^2 - 1 := by rw [g₀, g₁, gx₂] ring simpa [hx, hy] using he have hμy : μ + 2 * y₂ = 0 := (mul_eq_zero.mp hn).resolve_left hμ have hy₂ : y₂ = -μ/2 := by linarith have hc₀ : (1 - lam * μ) * y₀ = -1 := by rw [g₀] at h₀; nlinarith have hc₁ : (1 - lam * μ) * y₁ = 0 := by rw [g₁] at h₁; nlinarith have hc : 1 - lam * μ ≠ 0 := by intro h; rw [h] at hc₀; norm_num at hc₀ have hy₁ : y₁ = 0 := (mul_eq_zero.mp hc₁).resolve_left hc have hlam : lam ≠ 0 := by intro h rw [h] at h₂ simp only [zero_mul] at h₂ exact hμ (by linarith) have hdot : y₀ + 1 = lam * (μ + y₂) := by calc y₀ + 1 = (1 + y₀) * y₀ + y₁ * y₁ + y₂ * y₂ := by nlinarith [hy] _ = (lam * x₀) * y₀ + (lam * x₁) * y₁ + (lam * x₂) * y₂ := by rw [← h₀, ← h₁, ← h₂] _ = lam * (μ * (y₀^2 + y₁^2 + y₂^2) + y₂) := by rw [g₀, g₁, gx₂]; ring _ = lam * (μ + y₂) := by rw [hy]; ring have he : (lam * μ) * (2 * y₀ - 1) = 0 := by rw [g₀] at h₀ rw [hy₂] at hdot nlinarith only [h₀, hdot] have hy₀ : y₀ = 1/2 := by have := (mul_eq_zero.mp he).resolve_left (mul_ne_zero hlam hμ) linarith have hμsq : μ^2 = 3 := by rw [hy₀, hy₁, hy₂] at hy; nlinarith only [hy] refine ⟨hμsq, ?_, ?_, ?_, hy₀, hy₁, hy₂⟩ · rw [g₀, hy₀]; ring · rw [g₁, hy₁]; ring · rw [gx₂, hy₂]; nlinarith only [hμsq] lemma qzero_critical : MultiplierCritical qzero := by refine ⟨⟨?_, ?_⟩, 0, 0, ?_, ?_, ?_, ?_, ?_, ?_⟩ <;> norm_num [sqNorm, qzero, pole, east, Matrix.cons_val_two] lemma qplus_critical : MultiplierCritical qplus := by have hs : Real.sqrt 3 ^ 2 = 3 := Real.sq_sqrt (by norm_num) refine ⟨⟨?_, ?_⟩, Real.sqrt 3, Real.sqrt 3, ?_, ?_, ?_, ?_, ?_, ?_⟩ <;> norm_num [sqNorm, qplus, Matrix.cons_val_two] <;> nlinarith only [hs] lemma qminus_critical : MultiplierCritical qminus := by have hs : Real.sqrt 3 ^ 2 = 3 := Real.sq_sqrt (by norm_num) refine ⟨⟨?_, ?_⟩, -Real.sqrt 3, -Real.sqrt 3, ?_, ?_, ?_, ?_, ?_, ?_⟩ <;> norm_num [sqNorm, qminus, Matrix.cons_val_two] <;> nlinarith only [hs] /-- Exhaustiveness includes the isolated degenerate point. -/ theorem multiplierCritical_iff (q : Pair) : MultiplierCritical q ↔ q = qzero ∨ q = qplus ∨ q = qminus := by constructor · rintro ⟨⟨hx, hy⟩, lam, μ, h₀, h₁, h₂, g₀, g₁, g₂⟩ rcases classify_coordinates hx hy h₀ h₁ h₂ g₀ g₁ g₂ with h | h · left rcases h with ⟨hx₀, hx₁, hx₂, hy₀, hy₁, hy₂⟩ apply Prod.ext <;> ext i <;> fin_cases i <;> simp [qzero, pole, east, hx₀, hx₁, hx₂, hy₀, hy₁, hy₂] · rcases h with ⟨hμ, hx₀, hx₁, hx₂, hy₀, hy₁, hy₂⟩ have hs : Real.sqrt 3 ^ 2 = 3 := Real.sq_sqrt (by norm_num) have hm : μ = Real.sqrt 3 ∨ μ = -Real.sqrt 3 := sq_eq_sq_iff_eq_or_eq_neg.mp (hμ.trans hs.symm) rcases hm with hm | hm · right; left apply Prod.ext <;> ext i <;> fin_cases i <;> simp [qplus, hx₀, hx₁, hx₂, hy₀, hy₁, hy₂, hm] · right; right apply Prod.ext <;> ext i <;> fin_cases i <;> simp [qminus, hx₀, hx₁, hx₂, hy₀, hy₁, hy₂, hm] · rintro (rfl | rfl | rfl) · exact qzero_critical · exact qplus_critical · exact qminus_critical lemma qzero_ne_qplus : qzero ≠ qplus := by intro h have := congrArg (fun q : Pair => q.2 0) h norm_num [qzero, pole, east, qplus] at this lemma qzero_ne_qminus : qzero ≠ qminus := by intro h have := congrArg (fun q : Pair => q.2 0) h norm_num [qzero, pole, east, qminus] at this lemma qplus_ne_qminus : qplus ≠ qminus := by intro h have he := congrArg (fun q : Pair => q.1 0) h have hp : 0 < Real.sqrt 3 := Real.sqrt_pos.mpr (by norm_num) norm_num [qplus, qminus] at he linarith theorem multiplierCritical_set : {q : Pair | MultiplierCritical q} = {qzero, qplus, qminus} := by ext q simp only [Set.mem_ofPred_eq, Set.mem_insert_iff, Set.mem_singleton_iff, multiplierCritical_iff] /-- A finite cardinality assertion, not ncard's default zero for infinite sets. -/ theorem multiplierCritical_encard : {q : Pair | MultiplierCritical q}.encard = 3 := by rw [multiplierCritical_set, Set.encard_insert_of_notMem] · rw [Set.encard_pair qplus_ne_qminus] norm_num · simp [qzero_ne_qplus, qzero_ne_qminus] lemma f_qzero : f qzero = 0 := by norm_num [f, qzero, pole, east, Matrix.cons_val_two] lemma f_qplus : f qplus = 3 * Real.sqrt 3 / 2 := by norm_num [f, qplus, Matrix.cons_val_two]; ring lemma f_qminus : f qminus = -(3 * Real.sqrt 3 / 2) := by norm_num [f, qminus, Matrix.cons_val_two]; ring /-- Smoothness of the actual ambient polynomial, so its restriction is smooth. -/ theorem f_contDiff : ContDiff ℝ ∞ f := by unfold f; fun_prop open scoped Matrix lemma dot_self (x : R3) : x ⬝ᵥ x = sqNorm x := by simp [dotProduct, Fin.sum_univ_succ, sqNorm, pow_two] ring /-- The elementary tangent-annihilator lemma supplies (rather than assumes) the Lagrange multipliers. -/ lemma annihilator_sphere {x a : R3} (hx : sqNorm x = 1) : (∀ u : R3, x ⬝ᵥ u = 0 → a ⬝ᵥ u = 0) ↔ ∃ lam : ℝ, a = lam • x := by constructor · intro h let lam : ℝ := x ⬝ᵥ a let u : R3 := a - lam • x have hxu : x ⬝ᵥ u = 0 := by dsimp [u] rw [dotProduct_sub, dotProduct_smul, dot_self, hx] simp [lam] have hau : a ⬝ᵥ u = 0 := h u hxu have huu : u ⬝ᵥ u = 0 := by change (a - lam • x) ⬝ᵥ u = 0 rw [sub_dotProduct, smul_dotProduct, hau, hxu, smul_zero, sub_zero] exact ⟨lam, sub_eq_zero.mp (dotProduct_self_eq_zero.mp huu)⟩ · rintro ⟨lam, rfl⟩ u hu rw [smul_dotProduct, hu, smul_zero] def firstCoord (i : Fin 3) : Pair →L[ℝ] ℝ := (ContinuousLinearMap.proj i).comp (ContinuousLinearMap.fst ℝ R3 R3) def secondCoord (i : Fin 3) : Pair →L[ℝ] ℝ := (ContinuousLinearMap.proj i).comp (ContinuousLinearMap.snd ℝ R3 R3) /-- The ambient differential of the polynomial. -/ def df (q : Pair) : Pair →L[ℝ] ℝ := (1 + q.2 0) • firstCoord 0 + q.2 1 • firstCoord 1 + q.2 2 • firstCoord 2 + q.1 0 • secondCoord 0 + q.1 1 • secondCoord 1 + (q.1 2 - 1) • secondCoord 2 lemma df_apply (q r : Pair) : df q r = (east + q.2) ⬝ᵥ r.1 + (q.1 - pole) ⬝ᵥ r.2 := by simp [df, east, pole, firstCoord, secondCoord, dotProduct, Fin.sum_univ_succ] ring lemma f_hasFDerivAt (q : Pair) : HasFDerivAt f (df q) q := by have h₀ := (firstCoord 0).hasFDerivAt (x := q) have h₁ := (firstCoord 1).hasFDerivAt (x := q) have h₂ := (firstCoord 2).hasFDerivAt (x := q) have k₀ := (secondCoord 0).hasFDerivAt (x := q) have k₁ := (secondCoord 1).hasFDerivAt (x := q) have k₂ := (secondCoord 2).hasFDerivAt (x := q) convert! ((h₀.add (h₀.mul k₀)).add (h₁.mul k₁)).add ((h₂.sub_const 1).mul k₂) using 1 apply ContinuousLinearMap.ext intro r simp [df, firstCoord, secondCoord] ring /-- dh vanishes on the full constrained tangent of S² × S². -/ def TangentCritical (q : Pair) : Prop := q ∈ sphereProduct ∧ ∀ u v : R3, q.1 ⬝ᵥ u = 0 → q.2 ⬝ᵥ v = 0 → fderiv ℝ f q (u, v) = 0 lemma tangentCritical_iff_multiplierCritical (q : Pair) : TangentCritical q ↔ MultiplierCritical q := by have hd : fderiv ℝ f q = df q := (f_hasFDerivAt q).fderiv constructor · rintro ⟨hq, h⟩ have ha : ∀ u : R3, q.1 ⬝ᵥ u = 0 → (east + q.2) ⬝ᵥ u = 0 := by intro u hu have := h u 0 hu (dotProduct_zero _) simpa [hd, df_apply] using this have hb : ∀ v : R3, q.2 ⬝ᵥ v = 0 → (q.1 - pole) ⬝ᵥ v = 0 := by intro v hv have := h 0 v (dotProduct_zero _) hv simpa [hd, df_apply] using this obtain ⟨lam, hlam⟩ := (annihilator_sphere hq.1).mp ha obtain ⟨μ, hμ⟩ := (annihilator_sphere hq.2).mp hb refine ⟨hq, lam, μ, ?_, ?_, ?_, ?_, ?_, ?_⟩ · simpa [east] using congrFun hlam 0 · simpa [east] using congrFun hlam 1 · simpa [east] using congrFun hlam 2 · simpa [pole] using congrFun hμ 0 · simpa [pole] using congrFun hμ 1 · simpa [pole] using congrFun hμ 2 · rintro ⟨hq, lam, μ, h₀, h₁, h₂, g₀, g₁, g₂⟩ have ha : east + q.2 = lam • q.1 := by ext i; fin_cases i <;> simp [east, h₀, h₁, h₂] have hb : q.1 - pole = μ • q.2 := by ext i; fin_cases i <;> simp [pole, g₀, g₁, g₂] refine ⟨hq, ?_⟩ intro u v hu hv rw [hd, df_apply, ha, hb, smul_dotProduct, smul_dotProduct, hu, hv] simp /-- The complete critical-point calculation in constrained tangent form. -/ theorem tangentCritical_iff (q : Pair) : TangentCritical q ↔ q = qzero ∨ q = qplus ∨ q = qminus := by rw [tangentCritical_iff_multiplierCritical, multiplierCritical_iff] theorem tangentCritical_encard : {q : Pair | TangentCritical q}.encard = 3 := by simpa only [tangentCritical_iff_multiplierCritical] using multiplierCritical_encard end ThreePoints end ArnoldCounterexample end end OAI