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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 | import OAI.Analysis.C0Absorption.Shift
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
def baseRepresentation (γ : Label) : GeneratedRepresentation concreteBases γ where
operations := []
ordered := by simp
budget := by simpa only [List.map_nil,List.sum_nil] using half_pos (concreteBases.ell_pos (concreteBases.band γ))
theorem baseRepresentation_value (γ : Label) : (baseRepresentation γ).value=correction γ := by
funext s
change correctionBase γ (cubeRestrict (correctionCoordinates γ.level) s)=correction γ s
exact correctionBase_restrict γ s
theorem correction_mem_TClass (γ : Label) : correction γ∈TClass concreteBases 0 :=
⟨γ,baseRepresentation γ,Nat.zero_le _,(baseRepresentation_value γ).symm⟩
def correctionDualBound : ℝ≥0 := Classical.choose (TClass_pairing_bound concreteBases 0)
theorem correction_pairing_bound (γ : Label) (m : PreSpace (sourceClasses concreteBases) c0Origin) :
|pairing C0Ball (correction γ) m|≤correctionDualBound*‖m‖ :=
Classical.choose_spec (TClass_pairing_bound concreteBases 0) (correction γ) (correction_mem_TClass γ) m
def moleculeCorrection : Molecule C0Ball →ₗ[ℝ] BlockC0 :=
(Finsupp.linearCombination ℝ correctionC0).comp (MoleculeSpace C0Ball).subtype
theorem moleculeCorrection_apply (m : Molecule C0Ball) (γ : Label) :
moleculeCorrection m γ=pairing C0Ball (correction γ) m := by
classical
simp only [moleculeCorrection,pairing,LinearMap.comp_apply,Submodule.subtype_apply,
Finsupp.linearCombination_apply,Finsupp.sum]
induction (m.val.support) using Finset.induction_on with
| empty => simp
| @insert a s ha ih =>
simp only [Finset.sum_insert ha,ZeroAtInftyContinuousMap.add_apply,
ZeroAtInftyContinuousMap.smul_apply,smul_eq_mul,ih,correctionC0_apply]
def initialCorrection : PreSpace (sourceClasses concreteBases) c0Origin →L[ℝ] BlockC0 :=
(show PreSpace (sourceClasses concreteBases) c0Origin →ₗ[ℝ] BlockC0 from by unfold PreSpace; exact moleculeCorrection).mkContinuous
correctionDualBound (fun m => by
apply cfun_norm_le _ (mul_nonneg correctionDualBound.coe_nonneg (norm_nonneg m))
intro γ
change |moleculeCorrection m γ|≤_
exact (congrArg abs (moleculeCorrection_apply (show Molecule C0Ball from m) γ)).le.trans
(correction_pairing_bound γ m))
def correctionQ : ConcreteSpace →L[ℝ] BlockC0 :=
initialCorrection.extend (toTestSpace (sourceClasses concreteBases) c0Origin)
theorem correctionQ_coe (m : PreSpace (sourceClasses concreteBases) c0Origin) :
correctionQ (toTestSpace (sourceClasses concreteBases) c0Origin m)=moleculeCorrection m :=
ContinuousLinearMap.extend_eq _ UniformSpace.Completion.denseRange_coe
(UniformSpace.Completion.isUniformInducing_coe _) m
theorem correctionC0_zero : correctionC0 c0Origin=0 := by
ext γ
exact correction_zero γ
theorem correctionQ_zeta (s : C0Ball) :
correctionQ (zeta (sourceClasses concreteBases) c0Origin s)=correctionC0 s := by
rw [zeta,correctionQ_coe]
change moleculeCorrection (molecule s c0Origin)=correctionC0 s
simp only [moleculeCorrection,molecule,LinearMap.comp_apply,Submodule.subtype_apply,map_sub,
Finsupp.linearCombination_single,one_smul,correctionC0_zero,sub_zero]
def concreteLift : C0 → ConcreteSpace := radialLift (zeta (sourceClasses concreteBases) c0Origin)
theorem concreteLift_lipschitz : LipschitzWith 3 concreteLift :=
radialLift_lipschitz _ (zeta_lipschitz _ _) (zeta_base _ _)
theorem concreteLift_zero : concreteLift 0=0 := radialLift_zero _
theorem concreteLift_homogeneous {r : ℝ} (hr : 0≤r) (x : C0) : concreteLift (r • x)=r • concreteLift x :=
radialLift_homogeneous _ hr x
theorem correctionQ_concreteLift (x : C0) : correctionQ (concreteLift x)=recomputedShift x-inputBlocks x := by
rw [concreteLift,radialLift,map_smul,correctionQ_zeta]
change ‖x‖ • (frozenCorrectionL (stateWeights (normalizedBall x)) (normalize x))=_
rw [← map_smul,norm_smul_normalize]
change frozenCorrectionL (stateWeights (normalizedBall x)) x=
(inputBlocks x+frozenCorrectionL (stateWeights (normalizedBall x)) x)-inputBlocks x
abel
end
end C0Absorption
end OAI
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