Datasets:

roseking
/

openAI-math-dataset

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

	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