| import OAI.Geometry.Arnold.Obstruction |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators ContDiff |
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| open scoped BigOperators ContDiff |
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| namespace ArnoldCounterexample |
| namespace ThreePoints |
|
|
| abbrev R3 := Fin 3 → ℝ |
| abbrev Pair := R3 × R3 |
|
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| def sqNorm (x : R3) : ℝ := x 0 ^ 2 + x 1 ^ 2 + x 2 ^ 2 |
|
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| def sphereProduct : Set Pair := {q | sqNorm q.1 = 1 ∧ sqNorm q.2 = 1} |
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| 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 |
|
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| 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] |
|
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| 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 |
|
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| 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 |
|
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| 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 |
|
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| 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 |
|
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| 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) |
|
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| 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 |
|
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| end |
|
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| end OAI |
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