| import Mathlib | |
| import OAI.Analysis.BiholderTransport.Regularity.StationaryPullback | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter Manifold Bundle | |
| open scoped Topology ContDiff | |
| namespace WeakMTWTransport | |
| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M] | |
| [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M] | |
| [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) | |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] | |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| lemma divided_cost_difference_pullback {z : M} {p : TangentSpace 𝓘(ℝ,Model n) z} | |
| {t : ℝ} {g : E → TangentSpace 𝓘(ℝ,Model n) z} {a : E} | |
| (hp : p∈injectivityDomain z) (htp : t • p∈injectivityDomain z) (ht : t≠0) | |
| (hg : ContDiffAt ℝ 2 g a) (hg0 : g a=0) (k : E) : | |
| fderiv ℝ (fderiv ℝ (fun b => | |
| normalCost z (t • p) (g b)/t-normalCost z p (g b))) a k k = | |
| hessianValue z (t • p) (fderiv ℝ g a k)/t- | |
| hessianValue z p (fderiv ℝ g a k) := by | |
| let F := fun v => t⁻¹*normalCost z (t • p) v-normalCost z p v | |
| have hA := (normalCost_contDiffAt htp).of_le | |
| (m := 2) (ENat.natCast_le_of_coe_top_le_withTop le_rfl 2) | |
| have hB := (normalCost_contDiffAt hp).of_le | |
| (m := 2) (ENat.natCast_le_of_coe_top_le_withTop le_rfl 2) | |
| have hF : ContDiffAt ℝ 2 F 0 := (contDiffAt_const.mul hA).sub hB | |
| have hzero : fderiv ℝ F 0=0 := by | |
| ext v | |
| rw [fderiv_fun_sub ((hA.differentiableAt (by norm_num)).const_mul _) | |
| (hB.differentiableAt (by norm_num))] | |
| change (fderiv ℝ (t⁻¹ • normalCost z (t • p)) 0) v- | |
| fderiv ℝ (normalCost z p) 0 v=0 | |
| rw [fderiv_const_smul_field] | |
| simp only [Pi.smul_apply,smul_apply,smul_eq_mul, | |
| normalCost_fderiv_zero htp,normalCost_fderiv_zero hp,real_inner_smul_left] | |
| field_simp [ht] | |
| ring | |
| have HH := second_fderiv_comp_stationary (g := g) | |
| (by simpa only [hg0] using hF) hg (by simpa only [hg0] using hzero) k k | |
| rw [hg0,second_fderiv_sub (contDiffAt_const.mul hA) hB, | |
| second_fderiv_const_mul] at HH | |
| rw [hessianValue_eq_normalHessian htp,hessianValue_eq_normalHessian hp] | |
| simpa only [F,normalHessian,div_eq_mul_inv,mul_comm] using HH | |
| end WeakMTWTransport | |
| end | |
| end OAI | |