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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 | import OAI.Geometry.IsometricImmersion.Darboux.SixVariableLinearization
import Mathlib.Order.Preorder.Finite
namespace OAI
noncomputable section
open Set Function
open scoped ContDiff BigOperators Matrix
namespace SmoothLocal.HighEquation
open SmoothLocal.Geometry
def fullBlockPartition (ell : ℕ) (hell : 0 < ell) : OrderedFinpartition ell where
length := 1
partSize _ := ell
partSize_pos _ := hell
emb _ := id
emb_strictMono _ := strictMono_id
parts_strictMono := Subsingleton.strictMono _
disjoint := by
intro firstBlock _ secondBlock _ hne
exact (hne (Subsingleton.elim firstBlock secondBlock)).elim
cover i := ⟨0, i, rfl⟩
theorem ordered_full_size_length_one {ell : ℕ} (c : OrderedFinpartition ell)
{i : Fin c.length} (hi : c.partSize i = ell) : c.length = 1 := by
have hunique : ∀ j : Fin c.length, j = i := by
intro j
by_contra hji
have hb := ordered_two_partSizes_le c hji
have hj := c.partSize_pos j
rw [hi] at hb
omega
have hsub : Subsingleton (Fin c.length) :=
⟨fun a b => (hunique a).trans (hunique b).symm⟩
have hcard := Fintype.card_le_one_iff_subsingleton.mpr hsub
simp only [Fintype.card_fin] at hcard
have hi0 := i.isLt
omega
theorem ordered_length_one_eq_full {ell : ℕ} (hell : 0 < ell)
(c : OrderedFinpartition ell) (hc : c.length = 1) : c = fullBlockPartition ell hell := by
rcases c with ⟨length, size, hpos, emb, hmono, horder, hdis, hcover⟩
dsimp only at hc
subst length
have hsize0 : size 0 = ell := by
have hh := ordered_partSize_sum
(OrderedFinpartition.mk 1 size hpos emb hmono horder hdis hcover)
simpa only [Fin.sum_univ_one] using hh
have hsize : size = fun _ : Fin 1 => ell := by
funext i
have hi : i = 0 := Subsingleton.elim _ _
simpa only [hi] using hsize0
subst size
have hemb : emb = fun _ : Fin 1 => (id : Fin ell → Fin ell) := by
funext i
exact (hmono i).eq_id
subst emb
rfl
theorem ordered_nonfull_partSize_le {ell : ℕ} (hell : 0 < ell)
(c : OrderedFinpartition ell) (hc : c ≠ fullBlockPartition ell hell) (i : Fin c.length) :
c.partSize i ≤ ell - 1 := by
have hle := c.partSize_le i
have hne : c.partSize i ≠ ell := by
intro he
exact hc (ordered_length_one_eq_full hell c (ordered_full_size_length_one c he))
omega
def darbouxFaaTerm (g : MetricField) (z : Coord → ℝ) (x y : ℝ)
{ell : ℕ} (c : OrderedFinpartition ell) : ℝ :=
iteratedFDeriv ℝ c.length (sixVariableP g) (solutionJetCurve z x y)
(fun j => iteratedDeriv (c.partSize j) (solutionJetCurve z x) y)
def directLinearization (g : MetricField) (z u : Coord → ℝ) (p : Coord) : ℝ :=
heightPFirst g z 0 p * coordPartial 0 u p + heightPFirst g z 1 p * coordPartial 1 u p +
2 * hessianQuotient g z p * coordPartial 0 (coordPartial 1 u) p -
((hessianQuotient g z p)^2 + gaussianCurvature g p * darbouxG g z p) *
coordPartial 1 (coordPartial 1 u) p
theorem fullBlockPartition_term (g : MetricField) (z : Coord → ℝ) (x y : ℝ)
(ell : ℕ) (hell : 0 < ell) :
darbouxFaaTerm g z x y (fullBlockPartition ell hell) =
fderiv ℝ (sixVariableP g) (solutionJetCurve z x y)
(iteratedDeriv ell (solutionJetCurve z x) y) := by
exact iteratedFDeriv_one_apply (𝕜 := ℝ) (f := sixVariableP g)
(x := solutionJetCurve z x y) (fun _ : Fin 1 => iteratedDeriv ell (solutionJetCurve z x) y)
theorem fullBlockPartition_actual_linearization
{g : MetricField} {z : Coord → ℝ} {U : Set Coord} {x y : ℝ}
(hg : SmoothPositiveOn g U) (hU : IsOpen U) (hz : ContDiffOn ℝ ∞ z U)
(hp : (![x, y] : Coord) ∈ U) (hyy : covHessian g z ![x, y] 1 1 ≠ 0)
(ell : ℕ) (hell : 2 ≤ ell) :
darbouxFaaTerm g z x y (fullBlockPartition ell (by omega)) =
directLinearization g z (verticalJet z ell) ![x, y] := by
rw [fullBlockPartition_term, solutionJetCurve_iteratedDeriv hU hz hp ell hell]
change fderiv ℝ (sixVariableP g) (solutionJet z ![x, y]) _ = _
rw [sixVariableP_vertical_direction hg hU hp hyy _ (by rfl) (by rfl)]
rfl
theorem actualDarboux_extract_fullBlock
{g : MetricField} {z : Coord → ℝ} {U : Set Coord} {x y : ℝ}
(hg : SmoothPositiveOn g U) (hU : IsOpen U) (hz : ContDiffOn ℝ ∞ z U)
(hD : ∀ p ∈ U, (covHessian g z p).det = gaussianCurvature g p * heightEnergy g z p)
(hyy : ∀ p ∈ U, covHessian g z p 1 1 ≠ 0)
(hp : (![x, y] : Coord) ∈ U) (ell : ℕ) (hell : 2 ≤ ell) :
coordPartial 0 (coordPartial 0 (verticalJet z ell)) ![x, y] =
directLinearization g z (verticalJet z ell) ![x, y] +
∑ c ∈ (Finset.univ.erase (fullBlockPartition ell (by omega))), darbouxFaaTerm g z x y c := by
have heq := actualDarboux_faaExpansion_coord hg hU hz hD hyy hp ell
change coordPartial 0 (coordPartial 0 (verticalJet z ell)) ![x, y] =
∑ c : OrderedFinpartition ell, darbouxFaaTerm g z x y c at heq
have hsplit := (Finset.add_sum_erase (Finset.univ : Finset (OrderedFinpartition ell))
(darbouxFaaTerm g z x y) (Finset.mem_univ (fullBlockPartition ell (by omega)))).symm
rw [fullBlockPartition_actual_linearization hg hU hz hp (hyy _ hp) ell hell] at hsplit
exact heq.trans hsplit
end SmoothLocal.HighEquation
end
end OAI
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