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59d1c80 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 | import OAI.Geometry.IsometricImmersion.Taylor.ScalarChainExpansion
import OAI.Geometry.IsometricImmersion.Calculus.CoordinateJetNorm
import Mathlib.Data.List.Perm.Basic
import Mathlib.Data.List.Count
namespace OAI
noncomputable section
open Set Filter Function
open scoped ContDiff Topology BigOperators Matrix
namespace SmoothLocal.HighEquation
open SmoothLocal.Geometry
theorem orderedPartial_contDiffOn
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
{f : Coord → E} {U : Set Coord}
(hf : ContDiffOn ℝ ∞ f U) (hU : IsOpen U) (ds : List (Fin 2)) :
ContDiffOn ℝ ∞ (iteratedCoordPartial ds f) U := by
induction ds with
| nil => exact hf
| cons i ds ih => exact normed_coordPartial_contDiffOn ih hU i
theorem orderedPartial_append
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
(ds es : List (Fin 2)) (f : Coord → E) :
iteratedCoordPartial (ds ++ es) f = iteratedCoordPartial ds (iteratedCoordPartial es f) := by
induction ds with
| nil => rfl
| cons i ds ih => simp only [List.cons_append, iteratedCoordPartial, ih]
theorem orderedPartial_perm
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
{f : Coord → E} {U : Set Coord}
(hf : ContDiffOn ℝ ∞ f U) (hU : IsOpen U)
{ds es : List (Fin 2)} (hperm : ds.Perm es) :
∀ p ∈ U, iteratedCoordPartial ds f p = iteratedCoordPartial es f p := by
induction hperm with
| nil => intro p _; rfl
| cons i _perm ih =>
rename_i l1 l2
intro p hp
have he : iteratedCoordPartial l1 f =ᶠ[𝓝 p] iteratedCoordPartial l2 f := by
filter_upwards [hU.mem_nhds hp] with q hq
exact ih q hq
exact coordPartial_eq_of_eventuallyEq he i
| swap i j ds =>
intro p hp
exact normed_coordPartial_comm (orderedPartial_contDiffOn hf hU ds) hU hp j i
| trans _ _ ih₁ ih₂ =>
intro p hp
exact (ih₁ p hp).trans (ih₂ p hp)
theorem coordinateDirections_apply (ds : List (Fin 2)) (i : Fin ds.length) :
coordinateDirections ds i = Pi.single (ds.get i) (1 : ℝ) := by
induction ds with
| nil => exact Fin.elim0 i
| cons a ds ih =>
refine Fin.cases ?_ (fun j => ?_) i
· rfl
· exact ih j
theorem fullJet_coordinate_inputs_eq_ordered
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
{f : Coord → E} {U : Set Coord}
(hf : ContDiffOn ℝ ∞ f U) (hU : IsOpen U)
(n : ℕ) (d : Fin n → Fin 2) {p : Coord} (hp : p ∈ U) :
iteratedFDeriv ℝ n f p (fun i => Pi.single (d i) (1 : ℝ)) =
iteratedCoordPartial (List.ofFn d) f p := by
induction n generalizing p with
| zero => simp [iteratedCoordPartial]
| succ n ih =>
have hd := (hf.contDiffAt (hU.mem_nhds hp)).differentiableAt_iteratedFDeriv
(ENat.natCast_lt_of_coe_top_le_withTop le_rfl n)
rw [hd.iteratedFDeriv_succ_apply_left']
have he : (fun q => iteratedFDeriv ℝ n f q
(fun i => Pi.single (d i.succ) (1 : ℝ))) =ᶠ[𝓝 p]
iteratedCoordPartial (List.ofFn (fun i => d i.succ)) f := by
filter_upwards [hU.mem_nhds hp] with q hq
exact ih (fun i => d i.succ) hq
change fderiv ℝ (fun q => iteratedFDeriv ℝ n f q
(fun i => Pi.single (d i.succ) (1 : ℝ))) p (Pi.single (d 0) 1) = _
rw [he.fderiv_eq, List.ofFn_succ]
rfl
theorem state_fullJet_eval
{f : Coord → DarbouxState} {p : Coord} (hf : ContDiffAt ℝ ∞ f p)
(n : ℕ) (v : Fin n → Coord) (i : Fin 6) :
(iteratedFDeriv ℝ n f p v) i = iteratedFDeriv ℝ n (fun q => f q i) p v := by
let L : DarbouxState →L[ℝ] ℝ := ContinuousLinearMap.proj i
symm
change iteratedFDeriv ℝ n (L ∘ f) p v = L (iteratedFDeriv ℝ n f p v)
rw [L.iteratedFDeriv_comp_left hf (WithTop.coe_le_coe.mpr le_top)]
rfl
theorem fullJet_linear_zero
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
(L : Coord →L[ℝ] E) {n : ℕ} (hn : 2 ≤ n) (p : Coord) (v : Fin n → Coord) :
iteratedFDeriv ℝ n L p v = 0 := by
cases n with
| zero => omega
| succ n =>
have hn0 : n ≠ 0 := by omega
rw [iteratedFDeriv_succ_apply_right]
have he : (fun q => fderiv ℝ L q) = fun _ : Coord => L := by
funext q
exact L.fderiv
rw [he, iteratedFDeriv_const_of_ne hn0]
simp
theorem solutionJet_fullJet_ge_two
{z : Coord → ℝ} {U : Set Coord} {p : Coord}
(hU : IsOpen U) (hz : ContDiffOn ℝ ∞ z U) (hp : p ∈ U)
(n : ℕ) (hn : 2 ≤ n) (v : Fin n → Coord) :
iteratedFDeriv ℝ n (solutionJet z) p v =
![0, 0, iteratedFDeriv ℝ n (coordPartial 0 z) p v,
iteratedFDeriv ℝ n (coordPartial 1 z) p v,
iteratedFDeriv ℝ n (coordPartial 0 (coordPartial 1 z)) p v,
iteratedFDeriv ℝ n (coordPartial 1 (coordPartial 1 z)) p v] := by
have hJ := (solutionJet_contDiffOn hU hz).contDiffAt (hU.mem_nhds hp)
ext i
rw [state_fullJet_eval hJ n v i]
fin_cases i
· exact fullJet_linear_zero (ContinuousLinearMap.proj 0 : Coord →L[ℝ] ℝ) hn p v
· exact fullJet_linear_zero (ContinuousLinearMap.proj 1 : Coord →L[ℝ] ℝ) hn p v
· rfl
· rfl
· rfl
· rfl
theorem solutionJet_fullJet_one
{z : Coord → ℝ} {U : Set Coord} {p : Coord}
(hU : IsOpen U) (hz : ContDiffOn ℝ ∞ z U) (hp : p ∈ U) (v : Fin 1 → Coord) :
iteratedFDeriv ℝ 1 (solutionJet z) p v =
![(v 0) 0, (v 0) 1, iteratedFDeriv ℝ 1 (coordPartial 0 z) p v,
iteratedFDeriv ℝ 1 (coordPartial 1 z) p v,
iteratedFDeriv ℝ 1 (coordPartial 0 (coordPartial 1 z)) p v,
iteratedFDeriv ℝ 1 (coordPartial 1 (coordPartial 1 z)) p v] := by
have hJ := (solutionJet_contDiffOn hU hz).contDiffAt (hU.mem_nhds hp)
ext i
rw [state_fullJet_eval hJ 1 v i]
fin_cases i
· change iteratedFDeriv ℝ 1 (ContinuousLinearMap.proj 0 : Coord →L[ℝ] ℝ) p v = _
rw [iteratedFDeriv_one_apply, ContinuousLinearMap.fderiv]
rfl
· change iteratedFDeriv ℝ 1 (ContinuousLinearMap.proj 1 : Coord →L[ℝ] ℝ) p v = _
rw [iteratedFDeriv_one_apply, ContinuousLinearMap.fderiv]
rfl
· rfl
· rfl
· rfl
· rfl
theorem word_two_x_permutation (ds : List (Fin 2)) (hx : 2 ≤ ds.count 0) :
∃ es : List (Fin 2), ds.Perm (es ++ [0, 0]) := by
have h0 : (0 : Fin 2) ∈ ds := List.count_pos_iff.mp (by omega)
have h1 : (0 : Fin 2) ∈ ds.erase 0 := List.count_pos_iff.mp (by
rw [List.count_erase_self]
omega)
refine ⟨(ds.erase 0).erase 0, ?_⟩
have h := (List.perm_cons_erase h0).trans ((List.perm_cons_erase h1).cons 0)
exact h.trans (show ([0, 0] ++ (ds.erase 0).erase 0).Perm
((ds.erase 0).erase 0 ++ [0, 0]) from List.perm_append_comm)
theorem mixed_principal_word_orders (es : List (Fin 2)) :
(es ++ [0, 1]).length = (es ++ [0, 0]).length ∧
(es ++ [1, 1]).length = (es ++ [0, 0]).length ∧
(es ++ [0, 1]).count 0 < (es ++ [0, 0]).count 0 ∧
(es ++ [1, 1]).count 0 < (es ++ [0, 0]).count 0 := by
simp
end SmoothLocal.HighEquation
end
end OAI
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