| import OAI.Probability.InvariantIsing.Haar.HaarCurveBound |
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| namespace OAI |
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| /-! Finite-coordinate differentiable curves and their polynomial oscillation bounds. -/ |
| noncomputable section |
| open Matrix Set |
| open scoped BigOperators |
| namespace InvariantIsing |
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| structure HaarDifferentiableCurve (N : ℕ) where |
| value : ℝ → SpecialOrthogonal N |
| velocity : ℝ → Matrix (Fin N) (Fin N) ℝ |
| hasDeriv : ∀ t ∈ Icc (0:ℝ) 1, ∀ i j, |
| HasDerivAt (fun s => (value s : Matrix (Fin N) (Fin N) ℝ) i j) (velocity t i j) t |
| continuous_velocity : ∀ i j, ContinuousOn (fun t => velocity t i j) (Icc (0:ℝ) 1) |
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| namespace HaarDifferentiableCurve |
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| lemma continuous_entry {N : ℕ} (γ : HaarDifferentiableCurve N) (i j : Fin N) : |
| ContinuousOn (fun t => (γ.value t : Matrix (Fin N) (Fin N) ℝ) i j) (Icc (0:ℝ) 1) := |
| fun t ht => (γ.hasDeriv t ht i j).continuousAt.continuousWithinAt |
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| def constant {N : ℕ} (U : SpecialOrthogonal N) : HaarDifferentiableCurve N where |
| value := fun _ => U |
| velocity := fun _ => 0 |
| hasDeriv := fun t _ _ _ => hasDerivAt_const t _ |
| continuous_velocity := fun _ _ => continuousOn_const |
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| def mul {N : ℕ} (γ η : HaarDifferentiableCurve N) : HaarDifferentiableCurve N where |
| value := fun t => γ.value t*η.value t |
| velocity := fun t => γ.velocity t*(η.value t : Matrix (Fin N) (Fin N) ℝ)+ |
| (γ.value t : Matrix (Fin N) (Fin N) ℝ)*η.velocity t |
| hasDeriv := by |
| intro t ht i j |
| have hd := HasDerivAt.fun_sum (u := Finset.univ) fun k _ => |
| (γ.hasDeriv t ht i k).mul (η.hasDeriv t ht k j) |
| convert hd using 1 |
| · rfl |
| · simp only [Matrix.add_apply,Matrix.mul_apply,Finset.sum_add_distrib] |
| continuous_velocity := by |
| intro i j |
| simp only [Matrix.add_apply,Matrix.mul_apply] |
| exact (continuousOn_finsetSum _ fun k _ => |
| (γ.continuous_velocity i k).mul (η.continuous_entry k j)).add |
| (continuousOn_finsetSum _ fun k _ => |
| (γ.continuous_entry i k).mul (η.continuous_velocity k j)) |
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| lemma polynomial_bound {N : ℕ} (γ : HaarDifferentiableCurve N) : |
| ∃ K : ℝ, 0 ≤ K ∧ ∀ (p : MatrixPolynomial N) (δ : ℝ), 0 ≤ δ → |
| (∀ V : SpecialOrthogonal N, haarPolynomialValue (haarPolynomialGamma p p) V ≤ δ^2) → |
| |haarPolynomialValue p (γ.value 1)-haarPolynomialValue p (γ.value 0)| ≤ K*δ := by |
| let F := fun t => ∑ i : Fin N, ∑ j : Fin N, |
| |(γ.velocity t*(γ.value t : Matrix (Fin N) (Fin N) ℝ).transpose) i j| |
| have hF : ContinuousOn F (Icc (0:ℝ) 1) := by |
| apply continuousOn_finsetSum |
| intro i _ |
| apply continuousOn_finsetSum |
| intro j _ |
| apply ContinuousOn.abs |
| change ContinuousOn (fun t => ∑ k, γ.velocity t i k* |
| (γ.value t : Matrix (Fin N) (Fin N) ℝ) j k) _ |
| exact continuousOn_finsetSum _ fun k _ => |
| (γ.continuous_velocity i k).mul (γ.continuous_entry j k) |
| obtain ⟨C,hC⟩ := isCompact_Icc.bddAbove_image hF |
| refine ⟨max C 0,le_max_right _ _,?_⟩ |
| intro p δ hδ hp |
| have hk (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) : F t ≤ max C 0 := |
| (hC (mem_image_of_mem F ht)).trans (le_max_left _ _) |
| have hb := haarPolynomial_curve_bound p γ.value γ.velocity (max C 0) δ hδ γ.hasDeriv hk hp |
| exact hb.trans (by nlinarith [mul_nonneg (le_max_right C 0) hδ]) |
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| end HaarDifferentiableCurve |
| end InvariantIsing |
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| end |
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| end OAI |
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