| import OAI.Geometry.IsometricImmersion.Taylor.ActualDarbouxFaa |
| import Mathlib.Analysis.Calculus.ContDiff.Deriv |
| import Mathlib.Analysis.Calculus.FDeriv.Analytic |
| import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Function Filter |
| open scoped ContDiff Topology BigOperators |
|
|
| namespace SmoothLocal.HighEquation |
|
|
| structure ChainWord where |
| arity : ℕ |
| order : Fin arity → ℕ |
|
|
| def ChainWord.prepend (w : ChainWord) : ChainWord := |
| ⟨w.arity + 1, Fin.cons 1 w.order⟩ |
|
|
| def ChainWord.bump (w : ChainWord) (i : Fin w.arity) : ChainWord := |
| ⟨w.arity, Function.update w.order i (w.order i + 1)⟩ |
|
|
| def ChainWord.total (w : ChainWord) : ℕ := ∑ i, w.order i |
|
|
| def ChainWord.Positive (w : ChainWord) : Prop := ∀ i, 0 < w.order i |
|
|
| def ChainWord.Bounded (w : ChainWord) (n : ℕ) : Prop := ∀ i, w.order i ≤ n |
|
|
| def chainTerm {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (F : E → ℝ) (J : ℝ → E) (w : ChainWord) (t : ℝ) : ℝ := |
| iteratedFDeriv ℝ w.arity F (J t) (fun i => iteratedDeriv (w.order i) J t) |
|
|
| theorem iteratedDeriv_contDiffAt_infty |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {J : ℝ → E} {t : ℝ} (hJ : ContDiffAt ℝ ∞ J t) : |
| ∀ n, ContDiffAt ℝ ∞ (iteratedDeriv n J) t := by |
| intro n |
| induction n with |
| | zero => simpa using hJ |
| | succ n hn => |
| rw [iteratedDeriv_succ] |
| exact hn.derivWithin (by simp) |
|
|
| theorem iteratedDeriv_hasDerivAt_infty |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {J : ℝ → E} {t : ℝ} (hJ : ContDiffAt ℝ ∞ J t) (n : ℕ) : |
| HasDerivAt (iteratedDeriv n J) (iteratedDeriv (n + 1) J t) t := by |
| rw [iteratedDeriv_succ] |
| exact ((iteratedDeriv_contDiffAt_infty hJ n).differentiableAt (by simp)).hasDerivAt |
|
|
| theorem chainTerm_hasDerivAt |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {F : E → ℝ} {J : ℝ → E} {t : ℝ} |
| (hF : ContDiffAt ℝ ∞ F (J t)) (hJ : ContDiffAt ℝ ∞ J t) (w : ChainWord) : |
| HasDerivAt (chainTerm F J w) |
| (chainTerm F J w.prepend t + ∑ i, chainTerm F J (w.bump i) t) t := by |
| have hJ0 : HasDerivAt J (iteratedDeriv 1 J t) t := by |
| simpa using iteratedDeriv_hasDerivAt_infty hJ 0 |
| have houter := (hF.differentiableAt_iteratedFDeriv (m := w.arity) |
| (ENat.natCast_lt_of_coe_top_le_withTop le_rfl w.arity)).hasFDerivAt |
| |>.comp_hasDerivAt t hJ0 |
| have h := houter.hasFDerivAt.continuousMultilinearMap_apply |
| (fun i => (iteratedDeriv_hasDerivAt_infty hJ (w.order i)).hasFDerivAt) |
| |>.hasDerivAt |
| convert h using 1 <;> try rfl |
| simp only [chainTerm, ChainWord.prepend, ChainWord.bump, |
| add_apply, ContinuousLinearMap.comp_apply, |
| sum_apply, ContinuousLinearMap.toSpanSingleton_apply, one_smul, |
| ContinuousMultilinearMap.apply_apply] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro i hi |
| change (iteratedFDeriv ℝ w.arity F (J t)) |
| (fun j => iteratedDeriv (Function.update w.order i (w.order i + 1) j) J t) = |
| (iteratedFDeriv ℝ w.arity F (J t)) |
| (Function.update (fun j => iteratedDeriv (w.order j) J t) i (iteratedDeriv (w.order i + 1) J t)) |
| congr 1 |
| funext j |
| by_cases hj : j = i |
| · subst j; simp |
| · rw [Function.update_of_ne hj, Function.update_of_ne hj] |
|
|
| theorem ChainWord.prepend_positive {w : ChainWord} (hw : w.Positive) : |
| w.prepend.Positive := by |
| intro i |
| refine Fin.cases ?_ (fun j => ?_) i |
| · exact Nat.zero_lt_one |
| · exact hw j |
|
|
| theorem ChainWord.bump_positive {w : ChainWord} (hw : w.Positive) (i : Fin w.arity) : |
| (w.bump i).Positive := by |
| change ∀ j : Fin w.arity, 0 < Function.update w.order i (w.order i + 1) j |
| intro j |
| by_cases hj : j = i |
| · subst j; simp |
| · rw [Function.update_of_ne hj] |
| exact hw j |
|
|
| theorem ChainWord.prepend_total (w : ChainWord) : w.prepend.total = w.total + 1 := by |
| change (∑ j : Fin (w.arity + 1), Fin.cons 1 w.order j) = (∑ j : Fin w.arity, w.order j) + 1 |
| simp [Fin.sum_univ_succ, Nat.add_comm] |
|
|
| theorem ChainWord.bump_total (w : ChainWord) (i : Fin w.arity) : |
| (w.bump i).total = w.total + 1 := by |
| classical |
| have hi : i ∈ (Finset.univ : Finset (Fin w.arity)) := Finset.mem_univ i |
| have hsum := Finset.sum_update_of_mem hi w.order (w.order i + 1) |
| simp only [Finset.sdiff_singleton_eq_erase] at hsum |
| change (∑ j : Fin w.arity, Function.update w.order i (w.order i + 1) j) = |
| (∑ j : Fin w.arity, w.order j) + 1 |
| rw [hsum] |
| have he := Finset.sum_erase_add (Finset.univ : Finset (Fin w.arity)) w.order hi |
| omega |
|
|
| theorem ChainWord.prepend_bounded {w : ChainWord} {n : ℕ} |
| (hn : 1 ≤ n) (hw : w.Bounded n) : w.prepend.Bounded n := by |
| intro i |
| refine Fin.cases ?_ (fun j => ?_) i |
| · exact hn |
| · exact hw j |
|
|
| theorem ChainWord.bump_bounded {w : ChainWord} {n : ℕ} |
| (hw : w.Bounded n) (i : Fin w.arity) : (w.bump i).Bounded (n + 1) := by |
| change ∀ j : Fin w.arity, Function.update w.order i (w.order i + 1) j ≤ n + 1 |
| intro j |
| by_cases hj : j = i |
| · subst j |
| simpa [ChainWord.bump] using Nat.add_le_add_right (hw i) 1 |
| · rw [Function.update_of_ne hj] |
| exact Nat.le_succ_of_le (hw j) |
|
|
| theorem actualP_chainTerm_hasDerivAt |
| {g : SmoothLocal.Geometry.MetricField} {z : SmoothLocal.Geometry.Coord → ℝ} |
| {U : Set SmoothLocal.Geometry.Coord} {x y : ℝ} |
| (hg : SmoothLocal.Geometry.SmoothPositiveOn g U) (hU : IsOpen U) |
| (hz : ContDiffOn ℝ ∞ z U) (hp : (![x, y] : SmoothLocal.Geometry.Coord) ∈ U) |
| (hyy : SmoothLocal.Geometry.covHessian g z ![x, y] 1 1 ≠ 0) (w : ChainWord) : |
| HasDerivAt (chainTerm (sixVariableP g) (solutionJetCurve z x) w) |
| (chainTerm (sixVariableP g) (solutionJetCurve z x) w.prepend y + |
| ∑ i, chainTerm (sixVariableP g) (solutionJetCurve z x) (w.bump i) y) y := by |
| exact chainTerm_hasDerivAt |
| (sixVariableP_contDiffAt_solutionJet hg hU hp hyy) |
| (solutionJetCurve_contDiffAt hU hz hp) w |
|
|
| end SmoothLocal.HighEquation |
|
|
| end |
|
|
| end OAI |
|
|
