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9655094 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 | import OAI.Analysis.C0Absorption.FiniteShift
namespace OAI
open Set Filter Topology
open scoped NNReal BigOperators ZeroAtInfty
open NormedSpace
namespace C0Absorption
noncomputable section
open Set Filter Topology NormedSpace
open scoped NNReal BigOperators ZeroAtInfty
theorem mixed_vector_tendsto (s : C0Ball) (lev : Level) (b : Block lev) :
Tendsto (fun p : C0Ball × C0Ball => (mixedWeights p.1 p.2).vector lev b)
(nhds (s,s)) (nhds ((stateWeights s).vector lev b)) := by
classical
have h1 : Tendsto (fun p : C0Ball × C0Ball => blockVector lev p.1 b) (nhds (s,s)) (nhds (blockVector lev s b)) :=
((blockVector_continuous lev b).comp continuous_fst).continuousAt
have h2 : Tendsto (fun p : C0Ball × C0Ball => blockVector lev p.2 b) (nhds (s,s)) (nhds (blockVector lev s b)) :=
((blockVector_continuous lev b).comp continuous_snd).continuousAt
have he : (fun p : C0Ball × C0Ball => (mixedWeights p.1 p.2).vector lev b)=
(fun p : C0Ball × C0Ball => if localRadius lev p.2.val≤localRadius lev p.1.val then blockVector lev p.1 b else blockVector lev p.2 b) := by
funext p
dsimp only [mixedWeights,mixedState]
split_ifs <;> rfl
rw [he]
exact h1.if' h2
theorem mixed_row_tendsto (s : C0Ball) (lev : Level) (b : Block lev) :
Tendsto (fun p : C0Ball × C0Ball => (mixedWeights p.1 p.2).row lev b)
(nhds (s,s)) (nhds ((stateWeights s).row lev b)) := by
classical
have h1 : Tendsto (fun p : C0Ball × C0Ball => blockRow lev p.1 b) (nhds (s,s)) (nhds (blockRow lev s b)) :=
((blockRow_continuous lev b).comp continuous_fst).continuousAt
have h2 : Tendsto (fun p : C0Ball × C0Ball => blockRow lev p.2 b) (nhds (s,s)) (nhds (blockRow lev s b)) :=
((blockRow_continuous lev b).comp continuous_snd).continuousAt
have he : (fun p : C0Ball × C0Ball => (mixedWeights p.1 p.2).row lev b)=
(fun p : C0Ball × C0Ball => if localRadius lev p.2.val≤localRadius lev p.1.val then blockRow lev p.1 b else blockRow lev p.2 b) := by
funext p
dsimp only [mixedWeights,mixedState]
split_ifs <;> rfl
rw [he]
exact h1.if' h2
variable {N : ℕ}
theorem finiteRow_mixed_tendsto (s : C0Ball) (l : FLevel N) (x : FInput N) :
Tendsto (fun p : C0Ball × C0Ball => finiteRow (mixedWeights p.1 p.2) l x)
(nhds (s,s)) (nhds (finiteRow (stateWeights s) l x)) := by
exact tendsto_finsetSum _ (fun b _ => (mixed_row_tendsto s _ b).mul_const _)
theorem finitePrev_mixed_tendsto (s : C0Ball) (l : FLevel N) (x : FInput N) :
Tendsto (fun p : C0Ball × C0Ball => finitePrev (mixedWeights p.1 p.2) l x)
(nhds (s,s)) (nhds (finitePrev (stateWeights s) l x)) := by
unfold finitePrev
split_ifs
· exact tendsto_const_nhds
· exact finiteRow_mixed_tendsto s _ x
theorem finiteFrozen_mixed_apply_tendsto (s : C0Ball) (x : FInput N) :
Tendsto (fun p : C0Ball × C0Ball => finiteFrozen N (mixedWeights p.1 p.2) x)
(nhds (s,s)) (nhds (finiteFrozen N (stateWeights s) x)) := by
apply tendsto_pi_nhds.mpr
rintro (i|⟨l,b⟩)
· exact finiteRow_mixed_tendsto s (i,Fin.last N) x
· exact tendsto_const_nhds.add ((mixed_vector_tendsto s _ b).mul
((finitePrev_mixed_tendsto s l x).sub (finiteRow_mixed_tendsto s l x)))
theorem finiteFrozen_mixed_tendsto (N : ℕ) (s : C0Ball) :
Tendsto (fun p : C0Ball × C0Ball => finiteFrozen N (mixedWeights p.1 p.2))
(nhds (s,s)) (nhds (finiteFrozen N (stateWeights s))) := by
classical
exact tendsto_finiteOperator_of_columns _ _ (fun a => finiteFrozen_mixed_apply_tendsto s _)
theorem normalizedBall_continuousAt {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
{x : E} (hx : x≠0) : ContinuousAt (@normalizedBall E _ _ ) x := by
apply tendsto_subtype_rng.mpr
exact (continuousAt_id.norm.inv₀ (norm_ne_zero_iff.mpr hx)).smul continuousAt_id
theorem finite_normalized_continuousAt {x : FInput N} (hx : x≠0) :
ContinuousAt (fun y => normalizedBall (finiteEmbed N y)) x := by
have he : finiteEmbed N x≠0 := fun he => hx ((finiteEmbed N).injective (he.trans (map_zero _).symm))
exact (normalizedBall_continuousAt he).comp (finiteEmbed N).continuous.continuousAt
theorem finite_mixed_operator_near {x : FInput N} (hx : x≠0) :
∃ s : Set (FInput N), s∈nhds x ∧ ∀ y∈s, ∀ z∈s,
‖finiteFrozen N (mixedWeights (normalizedBall (finiteEmbed N y)) (normalizedBall (finiteEmbed N z)))-
finiteFrozen N (stateWeights (normalizedBall (finiteEmbed N x)))‖ < 1/20 := by
exact diagonal_neighborhood (fun y => normalizedBall (finiteEmbed N y))
(finite_normalized_continuousAt hx) (fun p => finiteFrozen N (mixedWeights p.1 p.2))
(finiteFrozen N (stateWeights (normalizedBall (finiteEmbed N x))))
(finiteFrozen_mixed_tendsto N _) (by norm_num)
theorem finiteRecomputed_local_onto {x : FInput N} (hx : x≠0) :
Set.range (finiteRecomputed N)∈nhds (finiteRecomputed N x) := by
let W := stateWeights (normalizedBall (finiteEmbed N x))
let S := finiteEquiv N W
obtain ⟨s,hs,hclose⟩ := finite_mixed_operator_near hx
apply range_mem_nhds_of_approx S s hs (1/10)
· have hh : ‖(S.symm : FInput N →L[ℝ] FInput N)‖₊≤5 := by
exact_mod_cast finiteEquiv_symm_norm N W
have hpos : 0<‖(S.symm : FInput N →L[ℝ] FInput N)‖₊ := by
exact S.nnnorm_symm_pos
rw [inv_eq_one_div,lt_div_iff₀ hpos]
calc (1/10 : ℝ≥0)*‖(S.symm : FInput N →L[ℝ] FInput N)‖₊ ≤ (1/10)*5 := mul_le_mul_of_nonneg_left hh (by positivity)
_ < 1 := by norm_num
· intro y hy z hz
let V := mixedWeights (normalizedBall (finiteEmbed N y)) (normalizedBall (finiteEmbed N z))
have he : ‖finiteRecomputed N y-finiteRecomputed N z-finiteFrozen N V (y-z)‖≤24*eta*dist y z :=
finiteRecomputed_mixed_approx y z
have hd : ‖(finiteFrozen N V-finiteFrozen N W) (y-z)‖≤(1/20)*dist y z := by
apply ((finiteFrozen N V-finiteFrozen N W).le_opNorm (y-z)).trans
rw [← dist_eq_norm y z]
exact mul_le_mul_of_nonneg_right (hclose y hy z hz).le dist_nonneg
have hid : finiteRecomputed N y-finiteRecomputed N z-finiteFrozen N W (y-z)=
(finiteRecomputed N y-finiteRecomputed N z-finiteFrozen N V (y-z))+
(finiteFrozen N V-finiteFrozen N W) (y-z) := by
rw [sub_apply]
abel
change ‖finiteRecomputed N y-finiteRecomputed N z-finiteFrozen N W (y-z)‖≤(1/10 : ℝ≥0)*‖y-z‖
rw [hid]
apply (norm_add_le _ _).trans
calc
_ ≤ 24*eta*dist y z+(1/20)*dist y z := add_le_add he hd
_ = (24*eta+1/20)*dist y z := by ring
_ ≤ (1/10)*dist y z := mul_le_mul_of_nonneg_right (by norm_num [eta]) dist_nonneg
_ = _ := by norm_num only [NNReal.coe_div,NNReal.coe_ofNat,NNReal.coe_one]; rw [dist_eq_norm]
end
end C0Absorption
end OAI
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