| import OAI.Analysis.DirectCrouzeix.AnalyticCharts |
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| namespace OAI |
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| universe u_135 u_136 u_137 |
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| noncomputable section |
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| open scoped Matrix Matrix.Norms.L2Operator Kronecker |
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| noncomputable section |
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| open MeasureTheory Set Filter Metric |
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| open scoped Topology Interval ENNReal NNReal ComplexConjugate |
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| noncomputable section |
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| open Filter Metric Set |
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| open scoped Topology ComplexConjugate |
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| noncomputable section |
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| open Set Filter Metric |
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| open scoped Topology ComplexConjugate |
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| noncomputable section |
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| open Set Filter Metric |
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| open scoped Topology ComplexConjugate |
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| namespace DirectCrouzeix.Geometry |
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| variable {ι : Type u_135} [Fintype ι] |
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| def expGradient (v : ι → ℂ) (b : ι → ℝ) (z : ℂ) : ℂ := |
| ∑ i, Real.exp (inner ℝ (v i) z+b i) • v i |
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| theorem expLevel_contDiff (v : ι → ℂ) (b : ι → ℝ) : ContDiff ℝ ⊤ (expLevel v b) := by |
| apply ContDiff.sum |
| intro i hi |
| exact ((innerSL ℝ (v i)).contDiff.add contDiff_const).exp |
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| theorem expLevel_hasFDerivAt (v : ι → ℂ) (b : ι → ℝ) (z : ℂ) : |
| HasFDerivAt (expLevel v b) (innerSL ℝ (expGradient v b z)) z := by |
| have hh := HasFDerivAt.fun_sum (x := z) (u := Finset.univ) (fun i _ => |
| (((innerSL ℝ (v i)).hasFDerivAt).add_const (b i)).exp) |
| convert! hh using 1 |
| ext w |
| simp only [expGradient,innerSL_apply_apply,sum_inner,real_inner_smul_left,sum_apply,smul_apply,smul_eq_mul] |
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| theorem expLevel_first_order (v : ι → ℂ) (b : ι → ℝ) (z w : ℂ) : |
| expLevel v b z + inner ℝ (expGradient v b z) (w-z) ≤ expLevel v b w := by |
| have hh (i : ι) : |
| Real.exp (inner ℝ (v i) z+b i) + Real.exp (inner ℝ (v i) z+b i)*inner ℝ (v i) (w-z) ≤ |
| Real.exp (inner ℝ (v i) w+b i) := by |
| have ht := mul_le_mul_of_nonneg_left (Real.add_one_le_exp (inner ℝ (v i) (w-z))) |
| (Real.exp_nonneg (inner ℝ (v i) z+b i)) |
| rw [← Real.exp_add] at ht |
| have he : inner ℝ (v i) z+b i+inner ℝ (v i) (w-z) = inner ℝ (v i) w+b i := by |
| rw [inner_sub_right]; ring |
| rw [he] at ht |
| linarith |
| have hs := Finset.sum_le_sum (fun i (_ : i ∈ Finset.univ) => hh i) |
| simpa only [expLevel,expGradient,sum_inner,real_inner_smul_left,Finset.sum_add_distrib] using hs |
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| theorem expGradient_ne_zero (v : ι → ℂ) (b : ι → ℝ) {p a : ℂ} |
| (hp : expLevel v b p = 1) (ha : expLevel v b a < 1) : expGradient v b p ≠ 0 := by |
| intro he |
| have hh := expLevel_first_order v b p a |
| rw [he,inner_zero_left,hp,add_zero] at hh |
| linarith |
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| theorem expLevel_convex (v : ι → ℂ) (b : ι → ℝ) : ConvexOn ℝ univ (expLevel v b) := by |
| refine ⟨convex_univ,?_⟩ |
| intro x hx y hy a c ha hc hac |
| have hh (i : ι) := convexOn_exp.2 (show inner ℝ (v i) x+b i ∈ univ from mem_univ _) |
| (show inner ℝ (v i) y+b i ∈ univ from mem_univ _) ha hc hac |
| have he (i : ι) : inner ℝ (v i) (a • x+c • y)+b i = |
| a*(inner ℝ (v i) x+b i)+c*(inner ℝ (v i) y+b i) := by |
| simp only [inner_add_right,inner_smul_right] |
| have hb := congrArg (fun u : ℝ => u*b i) hac |
| nlinarith |
| simp only [expLevel,he] |
| simpa [smul_eq_mul,Finset.sum_add_distrib,Finset.mul_sum] using |
| Finset.sum_le_sum (fun i (_ : i ∈ Finset.univ) => hh i) |
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| end DirectCrouzeix.Geometry |
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| namespace DirectCrouzeix.Geometry |
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| variable {ι : Type u_136} [Fintype ι] |
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| def expComplex (α β γ : ι → ℝ) (z : ℂ × ℂ) : ℂ := |
| ∑ i, Complex.exp ((γ i:ℂ)+(α i:ℂ)*z.1+(β i:ℂ)*z.2) |
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| theorem expComplex_analytic (α β γ : ι → ℝ) (z : ℂ × ℂ) : AnalyticAt ℂ (expComplex α β γ) z := by |
| apply Finset.analyticAt_fun_sum |
| intro i hi |
| exact ((analyticAt_const.add (analyticAt_const.mul analyticAt_fst)).add |
| (analyticAt_const.mul analyticAt_snd)).cexp |
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| theorem expComplex_zero (α β γ : ι → ℝ) : expComplex α β γ 0 = (∑ i, Real.exp (γ i):ℝ) := by |
| simp [expComplex,Complex.ofReal_exp] |
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| theorem expComplex_conj (α β γ : ι → ℝ) (z : ℂ × ℂ) : |
| expComplex α β γ (conj z.1,conj z.2) = conj (expComplex α β γ z) := by |
| simp only [expComplex,map_sum] |
| apply Finset.sum_congr rfl |
| intro i hi |
| rw [← Complex.exp_conj] |
| congr 1 |
| simp |
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| theorem expComplex_real (α β γ : ι → ℝ) (x y : ℝ) : |
| expComplex α β γ (x,y) = (∑ i, Real.exp (γ i+α i*x+β i*y):ℝ) := by |
| simp [expComplex,Complex.ofReal_exp] |
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| theorem expComplex_hasFDerivAt_zero (α β γ : ι → ℝ) : |
| HasFDerivAt (expComplex α β γ) |
| (((∑ i, Real.exp (γ i)*α i : ℝ):ℂ) • ContinuousLinearMap.fst ℂ ℂ ℂ + |
| ((∑ i, Real.exp (γ i)*β i : ℝ):ℂ) • ContinuousLinearMap.snd ℂ ℂ ℂ) 0 := by |
| have hh := HasFDerivAt.fun_sum (x := (0:ℂ × ℂ)) (u := Finset.univ) (fun i _ => |
| (((hasFDerivAt_const (γ i:ℂ) (0:ℂ × ℂ)).add |
| ((hasFDerivAt_fst (𝕜 := ℂ) (p := (0:ℂ × ℂ))).const_mul (α i:ℂ))).add |
| ((hasFDerivAt_snd (𝕜 := ℂ) (p := (0:ℂ × ℂ))).const_mul (β i:ℂ))).cexp) |
| convert! hh using 1 |
| apply ContinuousLinearMap.ext |
| intro z |
| simp [Complex.ofReal_exp,Finset.sum_mul,Finset.sum_add_distrib,mul_assoc] |
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| end DirectCrouzeix.Geometry |
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| namespace DirectCrouzeix.Geometry |
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| theorem closure_strict_level {f : ℂ → ℝ} (hc : Continuous f) |
| (hv : ConvexOn ℝ univ f) {a : ℂ} (ha : f a < 1) : |
| closure {z | f z < 1} = {z | f z ≤ 1} := by |
| apply Subset.antisymm |
| · apply closure_minimal |
| · intro z hz |
| exact (show f z < 1 from hz).le |
| · exact isClosed_le hc continuous_const |
| · intro z hz |
| let y := fun t : ℝ => (1-t) • z+t • a |
| have hy : Continuous y := by fun_prop |
| apply mem_closure_of_tendsto (b := 𝓝[>] (0:ℝ)) ((hy.tendsto' 0 z (by simp [y])).mono_left nhdsWithin_le_nhds) |
| filter_upwards [Ioo_mem_nhdsGT (by norm_num : (0:ℝ)<1)] with t ht |
| have hh := hv.2 (mem_univ z) (mem_univ a) (sub_nonneg.mpr (le_of_lt ht.2)) (le_of_lt ht.1) (sub_add_cancel 1 t) |
| change f (y t) ≤ (1-t)*f z+t*f a at hh |
| change f (y t) < 1 |
| have h1 := mul_le_mul_of_nonneg_left hz (sub_nonneg.mpr (le_of_lt ht.2)) |
| have h2 := mul_lt_mul_of_pos_left ha ht.1 |
| nlinarith |
| |
| end DirectCrouzeix.Geometry |
| |
| namespace DirectCrouzeix.Geometry |
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| variable {ι : Type u_137} [Fintype ι] |
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| end DirectCrouzeix.Geometry |
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| end |
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| end |
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| end |
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| end |
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| end |
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| end OAI |
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