| import Mathlib |
| import OAI.Analysis.BiholderTransport.CostGeometry.UniformFiniteRadialScale |
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| namespace OAI |
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| section |
| section |
| noncomputable section |
| open Set |
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| namespace WeakMTWTransport |
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| lemma finite_interval_mesh {K ε e : ℝ} (hK : 0 ≤ K) (hε : 0 < ε) |
| (he : e∈Ioo (0:ℝ) 1) : |
| ∃ N : ℕ, 0<N ∧ ∃ t : ℕ → ℝ, t 0=0 ∧ t N=1-e ∧ |
| (∀ k≤N, t k∈Icc (0:ℝ) (1-e)) ∧ |
| ∀ k<N, K*|t (k+1)-t k| ≤ ε := by |
| obtain ⟨N,hN⟩ := exists_nat_gt (K/ε+1) |
| have hn0 : (0:ℝ)<N := by |
| have := div_nonneg hK hε.le |
| linarith |
| have hNnat : 0<N := by exact_mod_cast hn0 |
| let t : ℕ → ℝ := fun k => (k:ℝ)/N*(1-e) |
| refine ⟨N,hNnat,t,by simp [t],by simp [t,ne_of_gt hn0],?_,?_⟩ |
| · intro k hk |
| have hr : (k:ℝ)/N ≤ 1 := (div_le_one hn0).mpr (by exact_mod_cast hk) |
| have hr0 : 0 ≤ (k:ℝ)/N := div_nonneg (Nat.cast_nonneg _) hn0.le |
| dsimp [t] |
| constructor |
| · exact mul_nonneg hr0 (by linarith [he.2]) |
| · have HH := mul_le_mul_of_nonneg_right hr (show 0 ≤ 1-e by linarith [he.2]) |
| simpa only [one_mul] using HH |
| · intro k hk |
| have hstep : t (k+1)-t k=(1-e)/N := by |
| dsimp [t] |
| rw [Nat.cast_add,Nat.cast_one] |
| ring |
| rw [hstep,abs_of_nonneg (div_nonneg (by linarith [he.2]) hn0.le)] |
| have hKN : K/N ≤ ε := by |
| rw [div_le_iff₀ hn0] |
| have HH := (div_lt_iff₀ hε).mp (show K/ε<(N:ℝ) by linarith [hN]) |
| linarith |
| have HH := mul_le_mul_of_nonneg_left (show (1-e)/N ≤ 1/N by |
| exact div_le_div_of_nonneg_right (by linarith [he.1]) hn0.le) hK |
| rw [mul_one_div] at HH |
| exact HH.trans hKN |
| end WeakMTWTransport |
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| end |
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| end |
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| section |
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| noncomputable section |
| open Set |
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| namespace WeakMTWTransport |
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| lemma uniform_finite_profile_comparison {C l u b D e K : ℝ} |
| (hC : 0<C) (hl : 0<l) (hu : 0<u) (hb : 0<b) (hD : 0<D) |
| (he : e∈Ioo (0:ℝ) 1) (heC : e*C ≤ l/32) (hK : 0 ≤ K) : |
| ∃ c : ℝ, 0<c ∧ ∀ (s : ℝ → ℝ) (f : ℝ → ℝ → ℝ), |
| (∀ t∈Icc (0:ℝ) 1, 0 ≤ s t) → s 1 ≤ D → |
| (∀ δ∈Ioc (0:ℝ) (1/2), ∀ t∈Icc (0:ℝ) 1, |
| l*δ/(s t+δ)-b*δ ≤ f δ t ∧ f δ t ≤ u*δ/(s t+δ)+b*δ) → |
| (∀ δ∈Ioc (0:ℝ) (1/2), ∀ x∈Icc (0:ℝ) (1-e), ∀ y∈Icc (0:ℝ) (1-e), |
| |f δ y-f δ x| ≤ K*|y-x|) → |
| (∀ δ∈Ioc (0:ℝ) (1/2), f δ (1-e) ≤ (1-e)*f δ 1+e*C) → |
| c*s 1 ≤ s 0 := by |
| classical |
| let ε := l/16 |
| have hε : 0<ε := by |
| dsimp [ε] |
| linarith only [heC, mul_pos he.1 hC] |
| obtain ⟨N,hN,t,ht0,htN,ht,hstep⟩ := finite_interval_mesh hK hε he |
| let L := 32*u/l+1 |
| have hL : 0<L := by dsimp [L]; positivity |
| have hR : 1 ≤ Real.exp L := (show 1 ≤ L+1 by linarith).trans (Real.add_one_le_exp L) |
| have hRu : u/Real.exp L ≤ l/32 := by |
| rw [div_le_iff₀ (Real.exp_pos L)] |
| have HH := Real.add_one_le_exp L |
| have HH2 := mul_le_mul_of_nonneg_left HH hl.le |
| have hcancel : l*(32*u/l)=32*u := by field_simp |
| dsimp [L] at HH2 |
| nlinarith only [HH2,hcancel,hl] |
| let η := min (1/2:ℝ) (l/(32*b)) |
| have hη : 0<η := lt_min (by norm_num) (div_pos hl (by positivity)) |
| obtain ⟨c,hc,HC⟩ := uniform_finite_radial_scale (N := N+1) hD hη hL |
| refine ⟨c,hc,?_⟩ |
| intro s f hs hsD hprof hlip hend |
| by_contra H |
| have hfail : s 0<c*s 1 := lt_of_not_ge H |
| have hs₀ := hs 0 (by constructor <;> norm_num) |
| have hs₁ : 0<s 1 := by nlinarith only [hfail,hs₀,hc] |
| let S := (Finset.range (N+1)).image (fun k => s (t k)) |
| have hSc : S.card ≤ N+1 := Finset.card_image_le.trans (by simp) |
| obtain ⟨δ,hd0,hdη,hdsmall,hdlarge,hdsep⟩ := HC S hSc (s 0) (s 1) hs₁ hsD hfail |
| have hd : δ∈Ioc (0:ℝ) (1/2) := ⟨hd0,hdη.trans (min_le_left _ _)⟩ |
| have hbg : b*δ ≤ l/32 := by |
| have HH := hdη.trans (min_le_right _ _) |
| have HH2 := (le_div_iff₀ (show 0<32*b by positivity)).mp HH |
| nlinarith only [HH2] |
| have hti (k : ℕ) (hk : k≤N) : t k∈Icc (0:ℝ) 1 := |
| ⟨(ht k hk).1,le_trans (ht k hk).2 (by linarith [he.1])⟩ |
| have hlarge_case (r : ℝ) (hr : r∈Icc (0:ℝ) 1) (hh : Real.exp L*s r ≤ δ) : |
| l/4 ≤ f δ r := by |
| have HH := radial_profile_small (hs r hr) hd0 hl.le hR hh hbg (hprof δ hd r hr).1 |
| linarith |
| have hsmall_case (r : ℝ) (hr : r∈Icc (0:ℝ) 1) (hh : δ*Real.exp L ≤ s r) : |
| f δ r ≤ ε := by |
| have HH := radial_profile_large (hs r hr) hd0 hu.le (Real.exp_pos L) hh hbg |
| (hprof δ hd r hr).2 |
| dsimp [ε] |
| linarith |
| have hfzero : l/4 ≤ f δ (t 0) := by |
| rw [ht0] |
| exact hlarge_case 0 (by constructor <;> norm_num) hdsmall |
| have hfend : f δ 1 ≤ ε := hsmall_case 1 (by constructor <;> norm_num) hdlarge |
| have hfbin (k : ℕ) (hk : k≤N) : f δ (t k) ≤ ε ∨ l/4 ≤ f δ (t k) := by |
| have hmem : s (t k)∈S := Finset.mem_image.mpr ⟨k,Finset.mem_range.mpr (by omega),rfl⟩ |
| rcases hdsep _ hmem with Hs|Hs |
| · exact Or.inr (hlarge_case (t k) (hti k hk) Hs) |
| · exact Or.inl (hsmall_case (t k) (hti k hk) Hs) |
| have hfstep (k : ℕ) (hk : k<N) : |f δ (t (k+1))-f δ (t k)| ≤ ε := |
| (hlip δ hd (t k) (ht k (by omega)) (t (k+1)) (ht (k+1) (by omega))).trans (hstep k hk) |
| have Hlarge := finite_spectral_gap_propagation (f := fun k => f δ (t k)) |
| (show 2*ε<l/4 by dsimp [ε]; linarith) hfbin hfstep hfzero |
| rw [htN] at Hlarge |
| have Hup := hend δ hd |
| have HH := mul_le_mul_of_nonneg_left hfend (show 0 ≤ 1-e by linarith [he.2]) |
| have HH2 := mul_nonneg hε.le he.1.le |
| dsimp [ε] at HH HH2 |
| nlinarith only [Hlarge,Hup,HH,HH2,heC,hl] |
| end WeakMTWTransport |
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| end |
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| end |
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| end |
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| end OAI |
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