openAI-math-dataset / lean /OAI /Analysis /BiholderTransport /LinearAlgebra /FrameDeterminantClosure.lean
| import Mathlib | |
| import OAI.Analysis.BiholderTransport.LinearAlgebra.ThreeDeterminantsClose | |
| import OAI.Analysis.BiholderTransport.Regularity.FrameDet | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Filter Manifold Bundle | |
| open scoped Topology ContDiff | |
| namespace WeakMTWTransport | |
| section FrameDeterminantClosure | |
| variable {n:ℕ} {M:Type*} [MetricSpace M] [CompactSpace M] [ConnectedSpace 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] | |
| lemma WeakMTW.frame_determinant_closure (hmtw:WeakMTW (n:=n) (M:=M)) | |
| {μ κ A χ K:ℝ} (hμ:0 < μ) (hκ:0 < κ) (hA:0 < A) (hχ:0 < χ) (hK:0 < K) : | |
| ∃B:ℝ,0 < B ∧ ∀(ι:Type) [Fintype ι] (a c:M) | |
| (r:ι → TangentSpace 𝓘(ℝ,Model n) a) (m:ι → ℝ) (q:TangentSpace 𝓘(ℝ,Model n) a), | |
| (∀j,0 ≤ m j) → (∑j,m j=1) → (∀j,r j∈minimizingVectors a) → | |
| (∑j,m j • r j=q) → ∀i (e:TangentSpace 𝓘(ℝ,Model n) a) (t D l lo:ℝ), | |
| μ ≤ m i → ‖e‖=1 → 0 < t → 0 < D → 0 < inner ℝ q e → | |
| κ*D ≤ (inner ℝ q e)^2 → ‖r i‖^2-‖q‖^2 ≤ A*D → r i=q+t • e → | |
| l∈Icc (1/2:ℝ) 1 → l<1 → lo∈Icc (1:ℝ) 2 → 1 < lo → | |
| ∀rc:TangentSpace 𝓘(ℝ,Model n) c,rc∈minimizingVectors c → | |
| reverseRay (⟨c,l • rc⟩:TangentBundle 𝓘(ℝ,Model n) M)=⟨a,q⟩ → | |
| ∀V W X:ℝ,0 < W → 0 < X → | |
| (chartFiberInverse a (extChartAt 𝓘(ℝ,Model n) a a)).toLinearMap.normDet^2*l^n* | |
| (expJacobian a q)^2*χ ≤ expJacobian c rc*V → | |
| expJacobian (n:=n) (reverseRay (⟨a,r i⟩:TangentBundle 𝓘(ℝ,Model n) M)).1 | |
| (lo⁻¹ • (reverseRay (⟨a,r i⟩:TangentBundle 𝓘(ℝ,Model n) M)).2)*W ≤ | |
| (chartFiberInverse a (extChartAt 𝓘(ℝ,Model n) a a)).toLinearMap.normDet^2*lo^n*K* | |
| (expJacobian a (r i))^2 → | |
| X*V ≤ W → X ≤ B := by | |
| have hc0:0 < (1/2:ℝ)^n*χ:=mul_pos (by positivity) hχ | |
| have hC0:0 < (2:ℝ)^n*K:=mul_pos (by positivity) hK | |
| obtain ⟨B,hB,HB⟩:=hmtw.uniform_three_determinant_closure hμ hκ hA hc0 | |
| (show (0:ℝ)<1 by norm_num) hC0 | |
| refine ⟨B,hB,?_⟩ | |
| intro ι _ a c r m q hmn hmt hr hbar i e t D l lo hmi he ht hD hqe hqD hgap hri | |
| hl hl1 hlo hlo1 rc hrc hrev V W X hW hX hdc hdo hXdet | |
| have hl0:0 < l:=lt_of_lt_of_le (by norm_num) hl.1 | |
| have hlo0:0 < lo:=zero_lt_one.trans hlo1 | |
| have hrev':reverseRay (⟨a,q⟩:TangentBundle 𝓘(ℝ,Model n) M)=⟨c,l • rc⟩:=by | |
| rw [←hrev,reverseRay_reverseRay] | |
| have hcontract:=contracted_minimizer_mem_injectivityDomain hrc hl0 hl1 | |
| have hqID:q∈injectivityDomain a:=by | |
| have H:=reverseRay_injectivityDomain (z:=⟨c,l • rc⟩) hcontract | |
| rwa [hrev] at H | |
| have hσ:=expJacobian_pos_of_injectivityDomain hqID | |
| have hσti:0 < expJacobian (n:=n) (reverseRay (⟨a,r i⟩:TangentBundle 𝓘(ℝ,Model n) M)).1 | |
| (lo⁻¹ • (reverseRay (⟨a,r i⟩:TangentBundle 𝓘(ℝ,Model n) M)).2):=by | |
| apply expJacobian_pos_of_injectivityDomain | |
| exact contracted_minimizer_mem_injectivityDomain (reverseRay_minimizing (hr i)) | |
| (inv_pos.mpr hlo0) ((inv_lt_one₀ hlo0).mpr hlo1) | |
| let f:ℝ:=(chartFiberInverse a (extChartAt 𝓘(ℝ,Model n) a a)).toLinearMap.normDet^2 | |
| have hf:0 < f:=sq_pos_of_pos (frameNormDet_pos (mem_extChartAt_target a)) | |
| have hlpow:(1/2:ℝ)^n ≤ l^n:=pow_le_pow_left₀ (by norm_num) hl.1 n | |
| have hlopow:lo^n ≤ (2:ℝ)^n:=pow_le_pow_left₀ hlo0.le hlo.2 n | |
| have hσc:0 < expJacobian c rc:=by | |
| have hp:0 < f*l^n*(expJacobian a q)^2*χ:=by positivity | |
| have hV:0 < expJacobian c rc*V:=hp.trans_le hdc | |
| exact lt_of_le_of_ne (expJacobian_nonneg c rc) (by | |
| intro hzero | |
| rw [←hzero,zero_mul] at hV | |
| exact (lt_irrefl (0:ℝ)) hV) | |
| have hσi:0 < expJacobian a (r i):=by | |
| have hp:0 < expJacobian (n:=n) (reverseRay (⟨a,r i⟩:TangentBundle 𝓘(ℝ,Model n) M)).1 | |
| (lo⁻¹ • (reverseRay (⟨a,r i⟩:TangentBundle 𝓘(ℝ,Model n) M)).2)*W:=mul_pos hσti hW | |
| have H:=hp.trans_le hdo | |
| exact lt_of_le_of_ne (expJacobian_nonneg a (r i)) (by | |
| intro hz | |
| rw [←hz,zero_pow (by decide),mul_zero] at H | |
| exact (lt_irrefl (0:ℝ)) H) | |
| have hlow:((1/2:ℝ)^n*χ)*(expJacobian a q)^2/expJacobian c rc ≤ V/f:=by | |
| apply (div_le_div_iff₀ hσc hf).mpr | |
| have H:=mul_le_mul_of_nonneg_right hlpow | |
| (show 0 ≤ f*(expJacobian a q)^2*χ by positivity) | |
| dsimp only [f] at H | |
| nlinarith only [H,hdc] | |
| have hupper:W/f≤((2:ℝ)^n*K)*(expJacobian a (r i))^2/ | |
| expJacobian (n:=n) (reverseRay (⟨a,r i⟩:TangentBundle 𝓘(ℝ,Model n) M)).1 | |
| (lo⁻¹ • (reverseRay (⟨a,r i⟩:TangentBundle 𝓘(ℝ,Model n) M)).2):=by | |
| apply (div_le_div_iff₀ hf hσti).mpr | |
| have H:=mul_le_mul_of_nonneg_right hlopow | |
| (show 0 ≤ f*K*(expJacobian a (r i))^2 by positivity) | |
| dsimp only [f] at H | |
| nlinarith only [H,hdo] | |
| have hratio:(1:ℝ)*X*(V/f)≤W/f:=by | |
| rw [one_mul,←mul_div_assoc] | |
| exact (div_le_div_iff_of_pos_right hf).mpr hXdet | |
| refine HB ι a r m q hmn hmt hr hbar.symm i e t D hmi he ht hD hqe hqD hgap hri l hl lo hlo | |
| ?_ hσ hσi ?_ hσti (V/f) (W/f) X hX ?_ hupper hratio | |
| · rw [hrev'] | |
| simpa only [smul_smul,inv_mul_cancel₀ hl0.ne',one_smul] using hrc | |
| · rw [hrev'] | |
| simpa only [smul_smul,inv_mul_cancel₀ hl0.ne',one_smul] using hσc | |
| · rw [hrev'] | |
| simpa only [smul_smul,inv_mul_cancel₀ hl0.ne',one_smul] using hlow | |
| end FrameDeterminantClosure | |
| end WeakMTWTransport | |
| end | |
| end | |
| end OAI | |