| import Mathlib |
| import OAI.Geometry.SmoothYau.Estimates.CoefficientElliptic |
| import OAI.Geometry.SmoothYau.Estimates.QuadraticInnerSpace |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace YauCounterexamples |
| section |
| open MeasureTheory TopologicalSpace |
| open scoped Distributions ContDiff |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| variable (K : Compacts E) {ι : Type*} [Fintype ι] (e : ι → E) |
|
|
| def coefficientTransport (X : ι → SmoothScalar E) (p : SmoothScalar E) : SmoothScalar E := |
| ∑ i, X i * SmoothScalar.directional (e i) p |
|
|
| def carlemanSymmetric (a : ι → ι → SmoothScalar E) (p : SmoothScalar E) (τ : ℝ) : |
| EllipticTest K →ₗ[ℝ] EllipticTest K := |
| testElliptic K e a + τ^2 • testMultiply K p |
|
|
| def carlemanSkew (X : ι → SmoothScalar E) (τ : ℝ) : |
| EllipticTest K →ₗ[ℝ] EllipticTest K := |
| (-τ) • testSkewTransport K e X |
|
|
| lemma testMultiplier_skew_pair (X : ι → SmoothScalar E) (p : SmoothScalar E) |
| (f : EllipticTest K) : |
| testPair K (testMultiply K p f) (testSkewTransport K e X f) = |
| -weightedPair K (coefficientTransport e X p) f f := by |
| have h (i : ι) := weightedPair_self_derivative K (p * X i) (e i) f |
| simp only [SmoothScalar.directional_mul, weightedPair_add_coeff] at h |
| simp only [testSkewTransport, LinearMap.add_apply, LinearMap.smul_apply, |
| testPair_add_right, testPair_smul_right, testTransport, LinearMap.sum_apply, |
| LinearMap.comp_apply, testPair_sum_right] |
| change 2 * (∑ i, weightedPair K p f (testMultiply K (X i) (testDerivative K (e i) f))) + |
| weightedPair K p f (testMultiply K (coefficientDivergence e X) f) = _ |
| simp only [weightedPair_mul, coefficientDivergence, Finset.mul_sum, |
| weightedPair_sum_coeff, coefficientTransport] |
| rw [← Finset.sum_add_distrib, ← Finset.sum_neg_distrib] |
| apply Finset.sum_congr rfl |
| intro i _ |
| rw [mul_comm (X i)] |
| linarith [h i] |
|
|
| lemma derivative_product_energy (a : ι → ι → SmoothScalar E) (X : ι → SmoothScalar E) |
| (f : EllipticTest K) : |
| (∑ i, ∑ j, ∑ k, weightedPair K (SmoothScalar.directional (e k) (a i j * X k)) |
| (testDerivative K (e j) f) (testDerivative K (e i) f)) = |
| (∑ i, ∑ j, ∑ k, weightedPair K (X k * SmoothScalar.directional (e k) (a i j)) |
| (testDerivative K (e j) f) (testDerivative K (e i) f)) + |
| ∑ i, ∑ j, weightedPair K (a i j * coefficientDivergence e X) |
| (testDerivative K (e j) f) (testDerivative K (e i) f) := by |
| simp only [SmoothScalar.directional_mul, weightedPair_add_coeff, |
| Finset.sum_add_distrib, coefficientDivergence, Finset.mul_sum, weightedPair_sum_coeff] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro i _ |
| apply Finset.sum_congr rfl |
| intro j _ |
| apply Finset.sum_congr rfl |
| intro k _ |
| rw [mul_comm] |
|
|
| theorem carleman_commutator (a : ι → ι → SmoothScalar E) |
| (ha : ∀ i j, a i j = a j i) (X : ι → SmoothScalar E) (p : SmoothScalar E) |
| (τ : ℝ) (f : EllipticTest K) : |
| 2 * testPair K (carlemanSymmetric K e a p τ f) (carlemanSkew K e X τ f) = |
| τ * (4 * (∑ i, ∑ j, ∑ k, |
| weightedPair K (a i j * SmoothScalar.directional (e i) (X k)) |
| (testDerivative K (e j) f) (testDerivative K (e k) f)) - |
| 2 * (∑ i, ∑ j, ∑ k, |
| weightedPair K (X k * SmoothScalar.directional (e k) (a i j)) |
| (testDerivative K (e j) f) (testDerivative K (e i) f)) - |
| weightedPair K (coefficientElliptic e a (coefficientDivergence e X)) f f) + |
| 2 * τ^3 * weightedPair K (coefficientTransport e X p) f f := by |
| have hT := testElliptic_transport_pair K e a ha X f |
| have hq := testElliptic_multiplier_pair K e a ha (coefficientDivergence e X) f |
| have hp := testMultiplier_skew_pair K e X p f |
| rw [derivative_product_energy K e a X f] at hT |
| simp only [carlemanSymmetric, carlemanSkew, LinearMap.add_apply, LinearMap.smul_apply, |
| testPair_smul_right, testPair_add_left, testPair_smul_left, hp] |
| simp only [testSkewTransport, LinearMap.add_apply, LinearMap.smul_apply, |
| testPair_add_right, testPair_smul_right] |
| linear_combination -τ * (2 * hT + hq) |
|
|
| lemma testPair_cauchy_shift (f g : EllipticTest K) (c : ℝ) : |
| -2 * c * testPair K f g - c^2 * testPair K g g ≤ testPair K f f := by |
| have h := testPair_nonneg K (f + c • g) |
| simp only [testPair_add_left, testPair_add_right, testPair_smul_left, |
| testPair_smul_right, testPair_comm K g f] at h |
| nlinarith |
|
|
|
|
| end |
|
|
| section |
| open MeasureTheory TopologicalSpace |
| open scoped Distributions ContDiff |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| variable {ι : Type*} [Fintype ι] (e : ι → E) |
|
|
| lemma SmoothScalar.sum_apply {κ : Type*} (s : Finset κ) (f : κ → SmoothScalar E) (x : E) : |
| (∑ i ∈ s, f i) x = ∑ i ∈ s, f i x := by |
| classical |
| induction s using Finset.induction_on with |
| | empty => simp only [Finset.sum_empty, Subalgebra.coe_zero, Pi.zero_apply] |
| | @insert i s hi ih => |
| simp only [Finset.sum_insert hi, Subalgebra.coe_add, Pi.add_apply, ih] |
|
|
| def coefficientGradient (a : ι → ι → SmoothScalar E) (φ : SmoothScalar E) : |
| ι → SmoothScalar E := fun i => ∑ j, a i j * SmoothScalar.directional (e j) φ |
|
|
| def coefficientNorm (a : ι → ι → SmoothScalar E) (φ : SmoothScalar E) : SmoothScalar E := |
| ∑ i, SmoothScalar.directional (e i) φ * coefficientGradient e a φ i |
|
|
| def convexifyPhase (ψ : SmoothScalar E) (H : ℝ) : SmoothScalar E := |
| ψ + (H/2) • (ψ*ψ) |
|
|
| lemma phase_derivative (ψ : SmoothScalar E) (H : ℝ) (v : E) : |
| SmoothScalar.directional v (convexifyPhase ψ H) = |
| (1 + H • ψ) * SmoothScalar.directional v ψ := by |
| simp only [convexifyPhase, map_add, map_smul, SmoothScalar.directional_mul] |
| apply Subtype.ext |
| ext x |
| simp only [Subalgebra.coe_add, Subalgebra.coe_mul, Subalgebra.coe_smul, Subalgebra.coe_one, |
| Pi.add_apply, Pi.mul_apply, Pi.smul_apply, Pi.one_apply, smul_eq_mul] |
| ring |
|
|
| lemma phase_gradient (a : ι → ι → SmoothScalar E) (ψ : SmoothScalar E) (H : ℝ) (i : ι) : |
| coefficientGradient e a (convexifyPhase ψ H) i = |
| (1 + H • ψ) * coefficientGradient e a ψ i := by |
| simp only [coefficientGradient, phase_derivative, Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro j _ |
| ring |
|
|
| lemma phase_norm (a : ι → ι → SmoothScalar E) (ψ : SmoothScalar E) (H : ℝ) : |
| coefficientNorm e a (convexifyPhase ψ H) = |
| (1 + H • ψ)^2 * coefficientNorm e a ψ := by |
| simp only [coefficientNorm, phase_derivative, phase_gradient, Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| ring |
|
|
| lemma SmoothScalar.directional_one (v : E) : SmoothScalar.directional v (1 : SmoothScalar E) = 0 := by |
| apply Subtype.ext |
| ext x |
| simp [SmoothScalar.directional_apply] |
|
|
| lemma phase_gradient_at (a : ι → ι → SmoothScalar E) (ψ : SmoothScalar E) (H : ℝ) |
| (x : E) (hx : ψ x = 0) (i : ι) : |
| coefficientGradient e a (convexifyPhase ψ H) i x = coefficientGradient e a ψ i x := by |
| rw [phase_gradient] |
| simp only [Subalgebra.coe_mul, Subalgebra.coe_add, Subalgebra.coe_one, Subalgebra.coe_smul, |
| Pi.mul_apply, Pi.add_apply, Pi.one_apply, Pi.smul_apply, smul_eq_mul, hx, |
| mul_zero, add_zero, one_mul] |
|
|
| lemma phase_norm_at (a : ι → ι → SmoothScalar E) (ψ : SmoothScalar E) (H : ℝ) |
| (x : E) (hx : ψ x = 0) : |
| coefficientNorm e a (convexifyPhase ψ H) x = coefficientNorm e a ψ x := by |
| rw [phase_norm] |
| simp only [Subalgebra.coe_mul, Subalgebra.coe_add, Subalgebra.coe_one, Subalgebra.coe_smul, |
| Subalgebra.coe_pow, Pi.mul_apply, Pi.add_apply, Pi.one_apply, Pi.smul_apply, |
| Pi.pow_apply, smul_eq_mul, hx, mul_zero, add_zero, one_pow, one_mul] |
|
|
| lemma phase_gradient_derivative_at (a : ι → ι → SmoothScalar E) (ψ : SmoothScalar E) (H : ℝ) |
| (x : E) (hx : ψ x = 0) (i j : ι) : |
| SmoothScalar.directional (e j) (coefficientGradient e a (convexifyPhase ψ H) i) x = |
| SmoothScalar.directional (e j) (coefficientGradient e a ψ i) x + |
| H * SmoothScalar.directional (e j) ψ x * coefficientGradient e a ψ i x := by |
| rw [phase_gradient, SmoothScalar.directional_mul, map_add, map_smul, |
| SmoothScalar.directional_one] |
| simp only [Subalgebra.coe_mul, Subalgebra.coe_add, Subalgebra.coe_one, Subalgebra.coe_smul, |
| Subalgebra.coe_zero, Pi.mul_apply, Pi.add_apply, Pi.one_apply, Pi.smul_apply, |
| Pi.zero_apply, smul_eq_mul, hx] |
| ring |
|
|
| lemma phase_transport_norm_at (a : ι → ι → SmoothScalar E) (ψ : SmoothScalar E) (H : ℝ) |
| (x : E) (hx : ψ x = 0) : |
| coefficientTransport e (coefficientGradient e a (convexifyPhase ψ H)) |
| (coefficientNorm e a (convexifyPhase ψ H)) x = |
| coefficientTransport e (coefficientGradient e a ψ) (coefficientNorm e a ψ) x + |
| 2*H*(coefficientNorm e a ψ x)^2 := by |
| have hd (i : ι) : |
| SmoothScalar.directional (e i) (coefficientNorm e a (convexifyPhase ψ H)) x = |
| SmoothScalar.directional (e i) (coefficientNorm e a ψ) x + |
| 2*H*SmoothScalar.directional (e i) ψ x * coefficientNorm e a ψ x := by |
| rw [phase_norm, pow_two, SmoothScalar.directional_mul, SmoothScalar.directional_mul, |
| map_add, map_smul, SmoothScalar.directional_one] |
| simp only [Subalgebra.coe_mul, Subalgebra.coe_add, Subalgebra.coe_one, Subalgebra.coe_smul, |
| Subalgebra.coe_zero, Pi.mul_apply, Pi.add_apply, Pi.one_apply, Pi.smul_apply, |
| Pi.zero_apply, smul_eq_mul, hx] |
| ring |
| simp only [coefficientTransport, SmoothScalar.sum_apply, |
| Subalgebra.coe_mul, Pi.mul_apply, phase_gradient_at e a ψ H x hx, hd, |
| mul_add, Finset.sum_add_distrib] |
| congr 1 |
| have hn : coefficientNorm e a ψ x = |
| ∑ i, SmoothScalar.directional (e i) ψ x * coefficientGradient e a ψ i x := by |
| simp only [coefficientNorm, SmoothScalar.sum_apply, Subalgebra.coe_mul, Pi.mul_apply] |
| calc |
| _ = 2*H*(∑ i, SmoothScalar.directional (e i) ψ x * coefficientGradient e a ψ i x) * |
| coefficientNorm e a ψ x := by |
| simp only [Finset.mul_sum, Finset.sum_mul] |
| apply Finset.sum_congr rfl |
| intro i _ |
| ring |
| _ = _ := by rw [← hn]; ring |
|
|
|
|
| end |
|
|
| section |
| open MeasureTheory TopologicalSpace |
| open scoped Distributions ContDiff |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| variable (K : Compacts E) {ι : Type*} [Fintype ι] (e : ι → E) |
|
|
| def exponentialWeight (φ : SmoothScalar E) (τ : ℝ) : SmoothScalar E := |
| ⟨fun x => Real.exp (τ * φ x), Real.contDiff_exp.comp (contDiff_const.mul (SmoothScalar.contDiff φ))⟩ |
| @[simp] lemma exponentialWeight_apply (φ : SmoothScalar E) (τ : ℝ) (x : E) : |
| exponentialWeight φ τ x = Real.exp (τ * φ x) := rfl |
|
|
| lemma exponentialWeight_derivative (φ : SmoothScalar E) (τ : ℝ) (v : E) : |
| SmoothScalar.directional v (exponentialWeight φ τ) = |
| τ • (exponentialWeight φ τ * SmoothScalar.directional v φ) := by |
| apply Subtype.ext |
| ext x |
| change fderiv ℝ (fun y => Real.exp (τ * φ y)) x v = _ |
| have hd := (((SmoothScalar.contDiff φ).differentiable (by simp) x).hasFDerivAt.const_mul τ).exp |
| rw [hd.fderiv] |
| simp only [smul_apply, smul_eq_mul, Subalgebra.coe_smul, Subalgebra.coe_mul, Pi.smul_apply, |
| Pi.mul_apply, SmoothScalar.directional_apply, exponentialWeight_apply] |
| ring |
|
|
| lemma testMultiply_one (f : EllipticTest K) : testMultiply K 1 f = f := by |
| ext x; simp [testMultiply_apply] |
|
|
| lemma testMultiply_exponential_inverse (φ : SmoothScalar E) (τ : ℝ) (f : EllipticTest K) : |
| testMultiply K (exponentialWeight φ (-τ)) (testMultiply K (exponentialWeight φ τ) f) = f := by |
| ext x |
| simp only [testMultiply_apply, exponentialWeight_apply, ← mul_assoc, ← Real.exp_add] |
| simp |
|
|
| lemma coefficientElliptic_exponential (a : ι → ι → SmoothScalar E) (φ : SmoothScalar E) (τ : ℝ) : |
| coefficientElliptic e a (exponentialWeight φ τ) = |
| exponentialWeight φ τ * |
| (τ • coefficientDivergence e (coefficientGradient e a φ) + |
| τ^2 • coefficientNorm e a φ) := by |
| have hg (i : ι) : (∑ j, a i j * SmoothScalar.directional (e j) (exponentialWeight φ τ)) = |
| τ • (exponentialWeight φ τ * coefficientGradient e a φ i) := by |
| apply Subtype.ext |
| ext x |
| simp only [exponentialWeight_derivative, coefficientGradient, SmoothScalar.sum_apply, |
| Subalgebra.coe_smul, Subalgebra.coe_mul, Pi.smul_apply, Pi.mul_apply, |
| smul_eq_mul, Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro j _ |
| ring |
| have he : coefficientElliptic e a (exponentialWeight φ τ) = |
| ∑ i, SmoothScalar.directional (e i) |
| (∑ j, a i j * SmoothScalar.directional (e j) (exponentialWeight φ τ)) := by |
| simp only [coefficientElliptic, map_sum] |
| rw [he] |
| simp_rw [hg] |
| simp only [map_smul, SmoothScalar.directional_mul, exponentialWeight_derivative, |
| coefficientDivergence, coefficientNorm, |
| Finset.sum_add_distrib, smul_add, Finset.smul_sum, mul_add, Finset.mul_sum, |
| smul_mul_assoc, mul_smul_comm, smul_smul] |
| simp only [← Finset.sum_add_distrib] |
| apply Finset.sum_congr rfl |
| intro i _ |
| apply Subtype.ext |
| ext x |
| simp only [Subalgebra.coe_add, Subalgebra.coe_smul, Subalgebra.coe_mul, |
| Pi.add_apply, Pi.smul_apply, Pi.mul_apply, smul_eq_mul] |
| ring |
|
|
| lemma testElliptic_product (a : ι → ι → SmoothScalar E) (ha : ∀ i j, a i j = a j i) |
| (b : SmoothScalar E) (f : EllipticTest K) : |
| testElliptic K e a (testMultiply K b f) = |
| testMultiply K b (testElliptic K e a f) + |
| (2:ℝ) • (∑ i, ∑ j, testMultiply K (a i j * SmoothScalar.directional (e i) b) |
| (testDerivative K (e j) f)) + testMultiply K (coefficientElliptic e a b) f := by |
| have hterm (i j : ι) : |
| testDerivative K (e i) (testMultiply K (a i j) (testDerivative K (e j) (testMultiply K b f))) = |
| testMultiply K b (testDerivative K (e i) (testMultiply K (a i j) (testDerivative K (e j) f))) + |
| testMultiply K (a i j * SmoothScalar.directional (e i) b) (testDerivative K (e j) f) + |
| testMultiply K (a i j * SmoothScalar.directional (e j) b) (testDerivative K (e i) f) + |
| testMultiply K (SmoothScalar.directional (e i) (a i j * SmoothScalar.directional (e j) b)) f := by |
| simp only [testProductRule, map_add, ← testMultiply_mul, |
| SmoothScalar.directional_mul, testMultiply_add] |
| ext x |
| simp only [FunLike.coe_add, Pi.add_apply, testMultiply_apply, |
| Subalgebra.coe_mul, Pi.mul_apply] |
| ring |
| have hcross : (∑ i, ∑ j, testMultiply K (a i j * SmoothScalar.directional (e j) b) |
| (testDerivative K (e i) f)) = |
| ∑ i, ∑ j, testMultiply K (a i j * SmoothScalar.directional (e i) b) |
| (testDerivative K (e j) f) := by |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro i _ |
| apply Finset.sum_congr rfl |
| intro j _ |
| rw [ha j i] |
| simp only [testElliptic, LinearMap.sum_apply, LinearMap.comp_apply, hterm, |
| Finset.sum_add_distrib, map_sum, coefficientElliptic, testMultiply_sum] |
| rw [hcross] |
| simp only [two_smul, add_assoc] |
|
|
| lemma testElliptic_exponential (a : ι → ι → SmoothScalar E) (ha : ∀ i j, a i j = a j i) |
| (φ : SmoothScalar E) (τ : ℝ) (f : EllipticTest K) : |
| testElliptic K e a (testMultiply K (exponentialWeight φ τ) f) = |
| testMultiply K (exponentialWeight φ τ) |
| (testElliptic K e a f + τ • testSkewTransport K e (coefficientGradient e a φ) f + |
| τ^2 • testMultiply K (coefficientNorm e a φ) f) := by |
| rw [testElliptic_product K e a ha, coefficientElliptic_exponential] |
| have hcross : (∑ i, ∑ j, testMultiply K |
| (a i j * SmoothScalar.directional (e i) (exponentialWeight φ τ)) (testDerivative K (e j) f)) = |
| τ • testMultiply K (exponentialWeight φ τ) (testTransport K e (coefficientGradient e a φ) f) := by |
| rw [Finset.sum_comm] |
| simp only [exponentialWeight_derivative, testTransport, coefficientGradient, |
| LinearMap.sum_apply, LinearMap.comp_apply, map_sum, Finset.smul_sum, |
| testMultiply_sum, ← testMultiply_mul, mul_smul_comm, testMultiply_smul] |
| apply Finset.sum_congr rfl |
| intro i _ |
| apply Finset.sum_congr rfl |
| intro j _ |
| rw [ha j i] |
| congr 2 |
| ring_nf |
| rw [hcross] |
| simp only [testMultiply_mul, testMultiply_add, testMultiply_smul, map_add, map_smul, |
| testSkewTransport, LinearMap.add_apply, LinearMap.smul_apply, smul_add, smul_smul] |
| module |
|
|
| lemma carleman_conjugation (a : ι → ι → SmoothScalar E) (ha : ∀ i j, a i j = a j i) |
| (φ : SmoothScalar E) (τ : ℝ) (u : EllipticTest K) : |
| carlemanSymmetric K e a (coefficientNorm e a φ) τ (testMultiply K (exponentialWeight φ τ) u) + |
| carlemanSkew K e (coefficientGradient e a φ) τ (testMultiply K (exponentialWeight φ τ) u) = |
| testMultiply K (exponentialWeight φ τ) (testElliptic K e a u) := by |
| have h := testElliptic_exponential K e a ha φ (-τ) (testMultiply K (exponentialWeight φ τ) u) |
| rw [testMultiply_exponential_inverse] at h |
| apply_fun testMultiply K (exponentialWeight φ τ) at h |
| have hi (f : EllipticTest K) : |
| testMultiply K (exponentialWeight φ τ) (testMultiply K (exponentialWeight φ (-τ)) f) = f := by |
| simpa only [neg_neg] using testMultiply_exponential_inverse K φ (-τ) f |
| rw [hi] at h |
| simpa only [carlemanSymmetric, carlemanSkew, LinearMap.add_apply, LinearMap.smul_apply, |
| neg_sq, add_assoc, add_left_comm, add_comm] using h.symm |
|
|
|
|
| end |
|
|
| open MeasureTheory TopologicalSpace Filter Set |
| open scoped Distributions ContDiff Topology |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
|
|
| def distanceSquare (x₀ : E) : SmoothScalar E := |
| ⟨fun x => ‖x-x₀‖^2, by |
| change ContDiff ℝ ∞ (fun x : E => ‖x-x₀‖^2) |
| exact (contDiff_id.sub (contDiff_const (c := x₀))).norm_sq ℝ⟩ |
| @[simp] lemma distanceSquare_apply (x₀ x : E) : distanceSquare x₀ x = ‖x-x₀‖^2 := rfl |
|
|
| lemma distanceSquare_derivative_center (x₀ v : E) : |
| SmoothScalar.directional v (distanceSquare x₀) x₀ = 0 := by |
| have hd := ((hasFDerivAt_id (𝕜 := ℝ) x₀).sub_const x₀).norm_sq |
| simp only [id_eq] at hd |
| change fderiv ℝ (fun x => ‖x-x₀‖^2) x₀ v = _ |
| rw [hd.fderiv] |
| simp |
|
|
| def tangentPhase (ψ : SmoothScalar E) (x₀ : E) : SmoothScalar E := ψ - distanceSquare x₀ |
| lemma tangentPhase_at (ψ : SmoothScalar E) (x₀ : E) (hψ : ψ x₀ = 0) : tangentPhase ψ x₀ x₀ = 0 := by |
| simp [tangentPhase, hψ] |
| lemma tangentPhase_derivative_center (ψ : SmoothScalar E) (x₀ v : E) : |
| SmoothScalar.directional v (tangentPhase ψ x₀) x₀ = SmoothScalar.directional v ψ x₀ := by |
| change SmoothScalar.directional v (ψ - distanceSquare x₀) x₀ = _ |
| rw [map_sub] |
| change SmoothScalar.directional v ψ x₀ - SmoothScalar.directional v (distanceSquare x₀) x₀ = _ |
| rw [distanceSquare_derivative_center, sub_zero] |
|
|
| lemma tangentPhase_annular_gap (ψ : SmoothScalar E) (x₀ x : E) (r H : ℝ) |
| (hr : 0 < r) (hψ : ψ x ≤ 0) (hd : r/2 ≤ dist x x₀) |
| (hsmall : -1 < H * tangentPhase ψ x₀ x) : |
| convexifyPhase (tangentPhase ψ x₀) H x ≤ -r^2/8 := by |
| have hnorm : r/2 ≤ ‖x-x₀‖ := by simpa only [dist_eq_norm] using hd |
| have hs : r^2/4 ≤ ‖x-x₀‖^2 := by nlinarith [norm_nonneg (x-x₀)] |
| have ht : tangentPhase ψ x₀ x ≤ -r^2/4 := by |
| change ψ x - ‖x-x₀‖^2 ≤ _ |
| linarith |
| have ht0 : tangentPhase ψ x₀ x ≤ 0 := ht.trans (by nlinarith [sq_nonneg r]) |
| have hmul := mul_nonpos_of_nonneg_of_nonpos (by linarith : 0 ≤ 1+H*tangentPhase ψ x₀ x) ht0 |
| change tangentPhase ψ x₀ x + (H/2)*(tangentPhase ψ x₀ x*tangentPhase ψ x₀ x) ≤ _ |
| nlinarith only [ht, hmul] |
|
|
|
|
|
|
| end YauCounterexamples |
| end |
|
|
| end OAI |
|
|
