| import OAI.Analysis.FreeGroupFactor.UnitaryEvaluation |
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| namespace OAI |
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| noncomputable section |
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| universe uA uι |
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| open Set Metric Filter |
| open scoped NNReal Topology BigOperators ContDiff |
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| open scoped Topology ContDiff |
| open Set Filter |
| namespace FreeGroupFactorMain |
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| theorem flat_local_of_factorial_bound (f : ℝ → ℝ) (hf : ContDiff ℝ ∞ f) |
| (B c : ℝ) (_hB : 0 ≤ B) (hc : 0 < c) |
| (hbound : ∀ r t, |iteratedDeriv r f t| ≤ B * c ^ r * (r.factorial : ℝ)) |
| (a : ℝ) (ha : ∀ r, iteratedDeriv (r + 1) f a = 0) |
| (x : ℝ) (hx : c * |x - a| < 1) : f x = f a := by |
| by_cases hax : a = x |
| · simp [hax] |
| have htaylor (n : ℕ) : taylorWithinEval f n (uIcc a x) a x = f a := by |
| induction n with |
| | zero => simp |
| | succ n ih => |
| rw [taylorWithinEval_succ, ih, |
| iteratedDerivWithin_eq_iteratedDeriv (uniqueDiffOn_uIcc hax) |
| ((hf.of_le (by exact_mod_cast (le_top : ((n + 1 : ℕ) : ℕ∞) ≤ ⊤))).contDiffAt) |
| (left_mem_uIcc), ha n] |
| simp |
| have hest (n : ℕ) : |f x - f a| ≤ B * (c * |x - a|) ^ (n + 1) := by |
| obtain ⟨y, _, hy⟩ := taylor_mean_remainder_lagrange_iteratedDeriv hax |
| ((hf.of_le (by exact_mod_cast (le_top : ((n + 1 : ℕ) : ℕ∞) ≤ ⊤))).contDiffOn) |
| rw [htaylor] at hy |
| have hfpos : (0 : ℝ) < ((n + 1).factorial : ℝ) := by positivity |
| calc |
| |f x - f a| = |iteratedDeriv (n + 1) f y| * |x - a| ^ (n + 1) / |
| ((n + 1).factorial : ℝ) := by rw [hy, abs_div, abs_mul, abs_pow, abs_of_pos hfpos] |
| _ ≤ (B * c ^ (n + 1) * ((n + 1).factorial : ℝ)) * |x - a| ^ (n + 1) / |
| ((n + 1).factorial : ℝ) := by gcongr; exact hbound _ _ |
| _ = B * (c * |x - a|) ^ (n + 1) := by rw [mul_pow]; field_simp |
| have hlim : Tendsto (fun n : ℕ => B * (c * |x - a|) ^ (n + 1)) atTop (nhds 0) := by |
| have hp := (tendsto_pow_atTop_nhds_zero_of_lt_one (mul_nonneg hc.le (abs_nonneg _)) hx) |
| simpa using (hp.comp (tendsto_add_atTop_nat 1)).const_mul B |
| have hz : |f x - f a| ≤ 0 := ge_of_tendsto hlim (Eventually.of_forall hest) |
| exact sub_eq_zero.mp (abs_nonpos_iff.mp hz) |
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| theorem flat_global_of_factorial_bound (f : ℝ → ℝ) (hf : ContDiff ℝ ∞ f) |
| (B c : ℝ) (hB : 0 ≤ B) (hc : 0 < c) |
| (hbound : ∀ r t, |iteratedDeriv r f t| ≤ B * c ^ r * (r.factorial : ℝ)) |
| (ha : ∀ r, iteratedDeriv (r + 1) f 0 = 0) (x : ℝ) : f x = f 0 := by |
| let Z : Set ℝ := {a | ∀ r, iteratedDeriv (r + 1) f a = 0} |
| have hclosed : IsClosed Z := by |
| change IsClosed {a | ∀ r, iteratedDeriv (r + 1) f a = 0} |
| simp only [ofPred_forall] |
| apply isClosed_iInter |
| intro r |
| exact isClosed_eq (hf.continuous_iteratedDeriv (r + 1) (by exact_mod_cast (le_top : ((r + 1 : ℕ) : ℕ∞) ≤ ⊤))) continuous_const |
| have hopen : IsOpen Z := by |
| apply isOpen_iff_mem_nhds.mpr |
| intro a ha |
| have hδ : 0 < c⁻¹ := inv_pos.mpr hc |
| apply Filter.mem_of_superset (Metric.ball_mem_nhds a hδ) |
| intro y hy |
| have heq : f =ᶠ[nhds y] (fun _ => f a) := by |
| filter_upwards [Metric.isOpen_ball.mem_nhds hy] with z hz |
| apply flat_local_of_factorial_bound f hf B c hB hc hbound a ha z |
| rw [Metric.mem_ball, Real.dist_eq] at hz |
| have := mul_lt_mul_of_pos_left hz hc |
| simpa [ne_of_gt hc] using this |
| intro r |
| rw [heq.iteratedDeriv_eq (r + 1)] |
| simp [iteratedDeriv_const] |
| have hall : Z = univ := (show IsClopen Z from ⟨hclosed, hopen⟩).eq_univ ⟨0, ha⟩ |
| have hderiv (a : ℝ) : deriv f a = 0 := by |
| have hz : a ∈ Z := hall ▸ mem_univ a |
| simpa using hz 0 |
| exact isOpen_univ.is_const_of_deriv_eq_zero isPreconnected_univ |
| (fun a _ => ((hf.of_le (by simp : (1 : ℕ∞ω) ≤ ∞)).differentiable (by simp)).differentiableOn a (mem_univ a)) |
| (fun a _ => hderiv a) (mem_univ x) (mem_univ 0) |
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| namespace NCPolynomial |
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| theorem iteratedDeriv_moment {ι : Type uι} {A : Type uA} [NormedRing A] [NormedAlgebra ℂ A] |
| (X : ℝ → ι → A) (V : ι → Expr ι) (L : A →L[ℝ] ℝ) |
| (hX : ∀ t i, HasDerivAt (fun s => X s i) (eval (X t) (V i)) t) |
| (p : Expr ι) (r : ℕ) : |
| iteratedDeriv r (fun t => L (eval (X t) p)) = |
| fun t => L (eval (X t) ((delta V)^[r] p)) := by |
| induction r with |
| | zero => rfl |
| | succ r ih => |
| rw [iteratedDeriv_succ, ih] |
| funext t |
| rw [Function.iterate_succ_apply'] |
| convert! (L.hasFDerivAt.comp_hasDerivAt t (hasDerivAt_eval X V t (hX t) ((delta V)^[r] p))).deriv using 1 |
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| theorem polynomial_moments {ι : Type uι} {A : Type uA} [NormedRing A] [NormedAlgebra ℂ A] |
| [NormOneClass A] (X : ℝ → ι → A) (V : ι → Expr ι) (L : A →L[ℝ] ℝ) |
| (hsmooth : ∀ i, ContDiff ℝ ∞ (fun t => X t i)) |
| (hX : ∀ t i, HasDerivAt (fun s => X s i) (eval (X t) (V i)) t) |
| (R : ℝ) (hR : 1 ≤ R) (hbounded : ∀ t i, ‖X t i‖ ≤ R) |
| (J : ℕ) (B : ℝ) (hB : 0 ≤ B) |
| (hdeg : ∀ i, degree (V i) ≤ J + 1) (hmass : ∀ i, mass (V i) ≤ B) |
| (htrace : ∀ p, L (eval (X 0) (delta V p)) = 0) |
| (p : Expr ι) (t : ℝ) : L (eval (X t) p) = L (eval (X 0) p) := by |
| let b := ‖L‖ * (mass p * R ^ degree p) |
| let c := B * ((degree p : ℝ) + J) * R ^ J + 1 |
| have hR0 : 0 ≤ R := le_trans zero_le_one hR |
| have hbase : 0 ≤ B * ((degree p : ℝ) + J) * R ^ J := by positivity |
| have hb : 0 ≤ b := mul_nonneg (norm_nonneg _) (mul_nonneg (mass_nonneg _) (pow_nonneg hR0 _)) |
| have hc : 0 < c := by dsimp [c]; linarith |
| apply flat_global_of_factorial_bound (fun t => L (eval (X t) p)) (L.contDiff.comp (smooth_eval X hsmooth p)) b c hb hc |
| · intro r u |
| rw [iteratedDeriv_moment X V L hX] |
| calc |
| |L (eval (X u) ((delta V)^[r] p))| ≤ ‖L‖ * ‖eval (X u) ((delta V)^[r] p)‖ := |
| L.le_opNorm _ |
| _ ≤ ‖L‖ * ((mass p * R ^ degree p) * |
| (B * ((degree p : ℝ) + J) * R ^ J) ^ r * (r.factorial : ℝ)) := by |
| gcongr |
| exact norm_eval_iterate (X u) R hR (hbounded u) V J B hB hdeg hmass p r |
| _ ≤ b * c ^ r * (r.factorial : ℝ) := by |
| dsimp [b] |
| calc |
| _ = (‖L‖ * (mass p * R ^ degree p)) * |
| (B * ((degree p : ℝ) + J) * R ^ J) ^ r * (r.factorial : ℝ) := by ring |
| _ ≤ _ := by |
| gcongr |
| dsimp [c]; linarith |
| · intro r |
| rw [iteratedDeriv_moment X V L hX, Function.iterate_succ_apply'] |
| exact htrace _ |
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| end NCPolynomial |
| end FreeGroupFactorMain |
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| end |
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| end OAI |
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