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dd14c26 | 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 Mathlib
import OAI.Probability.LogConcave.Sampling.SpatialTensorContinuous
namespace OAI
section
section
noncomputable section
namespace LogConcaveSampling.TensorEnergy
open Set Function
open scoped Classical BigOperators Matrix.Norms.L2Operator
structure SmoothTensorCurve (d : ℕ) (A : ℕ → ℝ) (R b : ℝ)
(X : ℝ × Point d → Point d) (V : ℝ → Point d → Point d) : Prop where
smooth : ContDiff ℝ (⊤:ℕ∞) X
initial : ∀y,X (0,y)=y
equation : ∀t∈Icc 0 b,∀y,HasDerivAt (fun u => X (u,y)) (V t (X (t,y))) t
velocity_smooth : ∀t∈Icc 0 b,ContDiff ℝ (⊤:ℕ∞) (V t)
velocity_bound : ∀n t,t∈Icc 0 b → ∀y,
AllSplitBound (multilinearTensor (iteratedFDeriv ℝ n (V t) y)) (A n/R^(n-1))
theorem smoothTensorCurve_bounds (A : ℕ → ℝ) (hA : ∀k,0≤A k) (n : ℕ) :
∃C : ℝ,0≤C ∧ ∀(d : ℕ) (R b : ℝ) (X : ℝ × Point d → Point d)
(V : ℝ → Point d → Point d), 0<R → R≤1 → 0≤b → b≤1 →
SmoothTensorCurve d A R b X V → ∀t∈Icc 0 b,∀y,
AllSplitBound (multilinearTensor (iteratedFDeriv ℝ n (fun z => X (t,z)) y))
(C/R^(n-1)) := by
induction n using Nat.strong_induction_on with
| h n ih =>
by_cases hn : n=0
· subst n
refine ⟨0,le_rfl,?_⟩
intro d R b X V hR hR1 hb hb1 hX t ht y
have he : (fun z => X (t,z))=(fun z => X (t,z)) := rfl
let : Subsingleton (Unit ⊕ Fin 0) := ⟨by intro a b; cases a with
| inl a => cases b with
| inl b => rfl
| inr b => exact Fin.elim0 b
| inr a => exact Fin.elim0 a⟩
exact allSplitBound_subsingleton _ _
have hn0 : 0<n := Nat.pos_of_ne_zero hn
choose C hC using fun k (hk : k<n) => ih k hk
let B (k : ℕ) : ℝ := if hk : k<n then C k hk else 0
have hB (k : ℕ) : 0≤B k := by
dsimp [B]
split_ifs with hk
· exact (hC k hk).1
· exact le_rfl
let F : ℝ := ∑c : OrderedFinpartition n,A c.length*∏i,B (c.partSize i)
let K : ℝ := (Fintype.card (OrderedFinpartition n):ℝ)*A 1
have hF : 0≤F := Finset.sum_nonneg (fun c _ => mul_nonneg (hA _) (Finset.prod_nonneg (fun i _ => hB _)))
have hK : 0≤K := mul_nonneg (Nat.cast_nonneg _) (hA 1)
have hG := gronwallBound_mono (δ:=1) (by norm_num) hF hK
have hG0 : 0≤gronwallBound 1 K F 1 := by
have hh := hG (show (0:ℝ)≤1 by norm_num)
rw [gronwallBound_x0] at hh
linarith
refine ⟨gronwallBound 1 K F 1,hG0,?_⟩
intro d R b X V hR hR1 hb hb1 hX t ht y
let T (u : ℝ) := multilinearTensor (iteratedFDeriv ℝ n (fun z => X (u,z)) y)
let D (u : ℝ) := multilinearTensor (iteratedFDeriv ℝ n (fun z => V u (X (u,z))) y)
have hpow : 0<R^(n-1) := pow_pos hR _
have hinit : AllSplitBound (T 0) (1/R^(n-1)) := by
have he : (fun z => X (0,z))=id := funext hX.initial
dsimp only [T]
rw [he]
exact (multilinearTensor_id_bound y).mono (by norm_num)
((le_div_iff₀ hpow).mpr (by simpa using (pow_le_one₀ hR.le hR1 : R^(n-1)≤1)))
have hd (u : ℝ) (hu : u∈Ico 0 b) : HasDerivWithinAt T (D u) (Ici u) u :=
(spatialTensor_time_deriv hX.smooth (V u) u (hX.equation u ⟨hu.1,hu.2.le⟩) y).hasDerivWithinAt
have hbound (u : ℝ) (hu : u∈Ico 0 b) :
AllSplitBound (D u) (K*‖splitBundle (T u)‖+F/R^(n-1)) := by
let z := ‖splitBundle (T u)‖
let N (k : ℕ) : ℝ := if k<n then B k/R^(k-1) else z
let M (k : ℕ) : ℝ := A k/R^(k-1)
have hM (k : ℕ) : 0≤M k := div_nonneg (hA _) (pow_nonneg hR.le _)
have hN (k : ℕ) : 0≤N k := by
dsimp [N,z]
split_ifs
· exact div_nonneg (hB _) (pow_nonneg hR.le _)
· exact norm_nonneg _
have hlow (k : ℕ) (hk : k≤n) : AllSplitBound
(multilinearTensor (iteratedFDeriv ℝ k (fun w => X (u,w)) y)) (N k) := by
by_cases hkn : k<n
· have hh := (hC k hkn).2 d R b X V hR hR1 hb hb1 hX u ⟨hu.1,hu.2.le⟩ y
simpa only [N,B,dite_eq_left hkn,ite_eq_left hkn] using hh
· have he : k=n := by omega
subst k
simpa only [N,lt_self_iff_false,ite_false,z,T] using allSplitBound_bundle (T u)
have hs : ContDiffAt ℝ (n:ℕ) (fun w => X (u,w)) y :=
(hX.smooth.comp (contDiff_const.prodMk contDiff_id)).contDiffAt.of_le (by exact_mod_cast (le_top : (n:ℕ∞)≤⊤))
have hv : ContDiffAt ℝ (n:ℕ) (V u) (X (u,y)) :=
(hX.velocity_smooth u ⟨hu.1,hu.2.le⟩).contDiffAt.of_le (by exact_mod_cast (le_top : (n:ℕ∞)≤⊤))
have hh := iteratedFDeriv_comp_allSplit y hs hv M N hM hN
(fun k hk => hX.velocity_bound k u ⟨hu.1,hu.2.le⟩ (X (u,y))) hlow
have hp := partition_majorant_linear hM
(fun k => div_nonneg (hB k) (pow_nonneg hR.le (k-1))) (norm_nonneg (splitBundle (T u)))
(fun k hk => show N k=B k/R^(k-1) from ite_eq_left hk)
(show N n=z from ite_eq_right (lt_irrefl n))
have he : (∑c : OrderedFinpartition n,M c.length*∏i,(B (c.partSize i)/R^(c.partSize i-1)))=
F/R^(n-1) := partition_sum_scale hn0 A B R
rw [he] at hp
have hp' : (∑c : OrderedFinpartition n,M c.length*∏i,N (c.partSize i))≤K*z+F/R^(n-1) := by
simpa only [M,Nat.sub_self,pow_zero,div_one,K,mul_assoc] using hp
exact hh.mono (Finset.sum_nonneg (fun c _ => mul_nonneg (hM _) (Finset.prod_nonneg (fun i _ => hN _)))) hp'
have hg := allSplit_gronwall (spatialTensor_continuous hX.smooth y).continuousOn hd
(div_nonneg (by norm_num) hpow.le) (div_nonneg hF hpow.le) hK hinit hbound ht
rw [gronwallBound_div] at hg
apply hg.mono (div_nonneg (by
have hh := hG (show (0:ℝ)≤t-0 by linarith [ht.1])
rw [gronwallBound_x0] at hh
linarith) hpow.le)
exact div_le_div_of_nonneg_right (hG (by linarith [ht.2])) hpow.le
end LogConcaveSampling.TensorEnergy
end
end
end
end OAI
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