openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Estimates /LargestBlockLinearization.lean
| 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 | |