| import Mathlib |
|
|
| namespace OAI |
| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped ContDiff |
| namespace ParabolicInterpolation |
| variable {V E : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] |
|
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| def difference : (n : ℕ) → (V → E) → (Fin n → V) → V → E |
| | 0,f,_,x => f x |
| | n+1,f,v,x => difference n f (Fin.tail v) (x+v 0)-difference n f (Fin.tail v) x |
|
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| def average : (n : ℕ) → (V → E) → (Fin n → V) → V → E |
| | 0,f,_,x => f x |
| | n+1,f,v,x => average n (fun y => ∫ t in (0:ℝ)..1, f (y+t • v 0)) (Fin.tail v) x |
| omit [CompleteSpace E] in |
| lemma continuous_average (n : ℕ) (f : V → E) (hf : Continuous f) (v : Fin n → V) : |
| Continuous (average n f v) := by |
| induction n generalizing f with |
| | zero => exact hf |
| | succ n ih => |
| apply ih |
| apply intervalIntegral.continuous_parametric_intervalIntegral_of_continuous' |
| exact hf.comp (by fun_prop) |
| omit [CompleteSpace E] in |
| lemma average_sub (n : ℕ) (f g : V → E) (hf : Continuous f) (hg : Continuous g) |
| (v : Fin n → V) (x : V) : |
| average n (fun y => f y-g y) v x=average n f v x-average n g v x := by |
| induction n generalizing f g with |
| | zero => rfl |
| | succ n ih => |
| change average n (fun y => ∫ t in (0:ℝ)..1, f (y+t • v 0)-g (y+t • v 0)) _ x=_ |
| have he (y : V) : (∫ t in (0:ℝ)..1, f (y+t • v 0)-g (y+t • v 0))= |
| (∫ t in (0:ℝ)..1, f (y+t • v 0))-(∫ t in (0:ℝ)..1, g (y+t • v 0)) := by |
| apply intervalIntegral.integral_sub |
| · exact (hf.comp (by fun_prop)).intervalIntegrable _ _ |
| · exact (hg.comp (by fun_prop)).intervalIntegrable _ _ |
| simp only [he] |
| apply ih |
| · apply intervalIntegral.continuous_parametric_intervalIntegral_of_continuous' |
| exact hf.comp (by fun_prop) |
| · apply intervalIntegral.continuous_parametric_intervalIntegral_of_continuous' |
| exact hg.comp (by fun_prop) |
| lemma average_const (n : ℕ) (c : E) (v : Fin n → V) (x : V) : |
| average n (fun _ => c) v x=c := by |
| induction n with |
| | zero => rfl |
| | succ n ih => simpa [average] using ih (Fin.tail v) |
| omit [CompleteSpace E] in |
| lemma average_translate (n : ℕ) (f : V → E) (v : Fin n → V) (x z : V) : |
| average n (fun y => f (y+z)) v x=average n f v (x+z) := by |
| induction n generalizing f x with |
| | zero => rfl |
| | succ n ih => |
| simp only [average] |
| have he : (fun y => ∫ t in (0:ℝ)..1, f (y+t • v 0+z))= |
| fun y => (fun w => ∫ t in (0:ℝ)..1, f (w+t • v 0)) (y+z) := by |
| funext y |
| congr 1 |
| funext t |
| congr 1 |
| abel |
| rw [he] |
| exact ih (fun y : V => ∫ t in (0:ℝ)..1, f (y+t • v 0)) (Fin.tail v) x |
| lemma average_jet_eq_difference (n : ℕ) (f : V → E) (hf : ContDiff ℝ n f) |
| (v : Fin n → V) (x : V) : |
| average n (fun y => iteratedFDeriv ℝ n f y v) v x=difference n f v x := by |
| induction n generalizing x with |
| | zero => rfl |
| | succ n ih => |
| let g : V → E := fun y => iteratedFDeriv ℝ n f y (Fin.tail v) |
| have hgd : Differentiable ℝ (iteratedFDeriv ℝ n f) := |
| hf.differentiable_iteratedFDeriv (by exact_mod_cast Nat.lt_succ_self n) |
| have hgc : Continuous g := hgd.continuous.eval_const _ |
| have he (y : V) : (∫ t in (0:ℝ)..1, iteratedFDeriv ℝ (n+1) f (y+t • v 0) v)=g (y+v 0)-g y := by |
| have hd (t : ℝ) : HasDerivAt (fun t => g (y+t • v 0)) |
| (iteratedFDeriv ℝ (n+1) f (y+t • v 0) v) t := by |
| rw [(hgd (y+t • v 0)).iteratedFDeriv_succ_apply_left'] |
| have hg : DifferentiableAt ℝ g (y+t • v 0) := |
| (hgd _).continuousMultilinear_apply_const _ |
| convert hg.hasFDerivAt.comp_hasDerivAt t ((hasDerivAt_id t).smul_const (v 0) |>.const_add y) using 1 |
| all_goals simp [Function.comp_def,g] |
| have hc0 : Continuous (fun z : V => iteratedFDeriv ℝ (n+1) f z v) := |
| (hf.continuous_iteratedFDeriv le_rfl).eval_const v |
| have hc1 : Continuous (fun t : ℝ => y+t • v 0) := by fun_prop |
| have hc : Continuous (fun t : ℝ => iteratedFDeriv ℝ (n+1) f (y+t • v 0) v) := hc0.comp hc1 |
| simpa [g] using intervalIntegral.integral_eq_sub_of_hasDerivAt |
| (fun t _ => hd t) (hc.intervalIntegrable 0 1) |
| change average n (fun y => ∫ t in (0:ℝ)..1, iteratedFDeriv ℝ (n+1) f (y+t • v 0) v) _ x=_ |
| simp only [he] |
| rw [average_sub n (fun y => g (y+v 0)) g (hgc.comp (continuous_id.add continuous_const)) hgc,average_translate] |
| rw [ih (hf.of_le (by exact_mod_cast Nat.le_succ n)),ih (hf.of_le (by exact_mod_cast Nat.le_succ n))] |
| rfl |
| omit [CompleteSpace E] in |
| lemma norm_average_le (n : ℕ) (f : V → E) (v : Fin n → V) (x : V) (C : ℝ) |
| (_hC : 0≤C) (hf : ∀ y, ‖y-x‖≤∑ i, ‖v i‖ → ‖f y‖≤C) : |
| ‖average n f v x‖≤C := by |
| induction n generalizing f with |
| | zero => exact hf x (by simp) |
| | succ n ih => |
| apply ih |
| intro y hy |
| have hi : ‖∫ t in (0:ℝ)..1, f (y+t • v 0)‖≤C*|1-0| := |
| intervalIntegral.norm_integral_le_of_norm_le_const (fun t ht => by |
| apply hf |
| have ht' : t∈Icc (0:ℝ) 1 := by |
| simp only [uIoc_of_le zero_le_one] at ht |
| exact ⟨ht.1.le,ht.2⟩ |
| calc |
| ‖y+t • v 0-x‖=‖(y-x)+t • v 0‖ := by congr 1; abel |
| _≤‖y-x‖+‖t • v 0‖ := norm_add_le _ _ |
| _≤(∑ i, ‖Fin.tail v i‖)+‖v 0‖ := by |
| gcongr |
| rw [norm_smul,Real.norm_of_nonneg ht'.1] |
| exact mul_le_of_le_one_left (norm_nonneg _) ht'.2 |
| _=∑ i, ‖v i‖ := by rw [Fin.sum_univ_succ]; simp [Fin.tail]; ring) |
| simpa using hi |
| omit [NormedSpace ℝ V] [NormedSpace ℝ E] [CompleteSpace E] in |
| lemma norm_difference_le (n : ℕ) (f : V → E) (v : Fin n → V) (x : V) (U : ℝ) |
| (_hU : 0≤U) (hf : ∀ y, ‖y-x‖≤∑ i, ‖v i‖ → ‖f y‖≤U) : |
| ‖difference n f v x‖≤2^n*U := by |
| induction n generalizing x with |
| | zero => simpa [difference] using hf x (by simp) |
| | succ n ih => |
| have hsum : (∑ i : Fin n, ‖Fin.tail v i‖)+‖v 0‖=∑ i, ‖v i‖ := by |
| rw [Fin.sum_univ_succ]; simp [Fin.tail]; ring |
| have h1 := ih (Fin.tail v) (x+v 0) (fun y hy => hf y (by |
| calc |
| ‖y-x‖=‖(y-(x+v 0))+v 0‖ := by congr 1; abel |
| _≤‖y-(x+v 0)‖+‖v 0‖ := norm_add_le _ _ |
| _≤(∑ i : Fin n, ‖Fin.tail v i‖)+‖v 0‖ := add_le_add hy le_rfl |
| _=_ := hsum)) |
| have h2 := ih (Fin.tail v) x (fun y hy => hf y (by |
| rw [←hsum]; exact hy.trans (le_add_of_nonneg_right (norm_nonneg _)))) |
| exact (norm_sub_le _ _).trans (by change _≤2^(n+1)*U; rw [pow_succ]; linarith) |
| end ParabolicInterpolation |
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| end |
|
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| end OAI |
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