Datasets:

roseking
/

openAI-math-dataset

import OAI.Probability.ProjectionMoments.Seeded

namespace OAI

noncomputable section

open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace
namespace NoiselessRegression
namespace Learner
variable {d M T : ℕ} {Ω : Type*} [MeasurableSpace Ω]

lemma uniformSuccess_le_sum_fixed (L : Learner d M T Ω) (ρ : Measure Ω) (ε : ℝ) :
    L.uniformSuccess ρ ε ≤ ∑ t : Fin (T+1), L.uniformFixedSuccess ρ ε t := by
  classical
  let μ := (uniformSphere d).prod (ρ.prod (sampleLaw d T))
  have hs : {z : UnitSphere d × (Ω × Samples d T) |
      angularError z.1 (L.estimate z.1 z.2.1 z.2.2) ≤ ε} =
      ⋃ t : Fin (T+1), {z | (L.terminal z.1 z.2.1 z.2.2).1 = t ∧
        angularError z.1 (L.estimate z.1 z.2.1 z.2.2) ≤ ε} := by
    ext z
    simp only [Set.mem_ofPred_eq,Set.mem_iUnion]
    constructor
    · intro hz
      obtain ⟨t,ht,he⟩ := L.terminal_forced_witness z.1 z.2.1 z.2.2
      exact ⟨⟨t,by omega⟩,by simpa [he] using hz⟩
    · rintro ⟨t,ht,hz⟩
      exact hz
  change μ _ ≤ ∑ t : Fin (T+1), μ _
  rw [hs]
  exact (measure_iUnion_le _).trans_eq (tsum_fintype _)

end Learner
namespace FiniteKernelLearner

section
universe u

theorem cap_total_success :
    ∃ C : ℝ, 0 < C ∧ ∀ d M T : ℕ, 32 ≤ d → ∀ ε : ℝ,
      0 < ε → ε ≤ 1/10 →
      ∀ (Ξ : Type u) [MeasurableSpace Ξ] (ρ : Measure Ξ), IsProbabilityMeasure ρ →
      ∀ K : Ξ → FiniteKernelLearner d M T, (∀ᵐ ξ ∂ρ, (K ξ).CompletedRules) →
      let k := d/16
      let α : ℝ := ((d:ℝ)-1)/2
      seededSuccess K ρ ε ≤ ENNReal.ofReal
        (((T:ℝ)+1)*Real.exp (C*d)*(ε/2)^α*
          (Real.exp (C*d)*(2:ℝ)^((M:ℝ)/k))^((T+k-1)/k)) := by
  obtain ⟨C,hC,h⟩ := Learner.cap_fixed_stop.{0}
  refine ⟨C,hC,?_⟩
  intro d M T hd ε hε hε10 Ξ _ ρ hρ K hK k α
  let : IsProbabilityMeasure ρ := hρ
  apply seed_average_bound
  filter_upwards [hK] with ξ hξ
  obtain ⟨ν,L,hν,hB,hL,he,_⟩ := (K ξ).exists_equivalent_tape hξ (by omega)
  rw [←he ε]
  have hs := L.uniformSuccess_le_sum_fixed ν ε
  have hbase : 1 ≤ Real.exp (C*d)*(2:ℝ)^((M:ℝ)/k) :=
    one_le_mul_of_one_le_of_one_le (Real.one_le_exp (by positivity))
      (Real.one_le_rpow (by norm_num) (by positivity))
  apply hs.trans
  calc
    _ ≤ ∑ _t : Fin (T+1), ENNReal.ofReal
        (Real.exp (C*d)*(ε/2)^α*(Real.exp (C*d)*(2:ℝ)^((M:ℝ)/k))^((T+k-1)/k)) := by
      apply Finset.sum_le_sum
      intro t _
      apply (h d M T hd ε hε hε10 (Tape M T) ν hν L hL
        (L.measurable_terminal hB).fst.aemeasurable t (by omega)).trans
      apply ENNReal.ofReal_le_ofReal
      apply mul_le_mul_of_nonneg_left _ (by positivity)
      exact pow_le_pow_right₀ hbase (Nat.div_le_div_right (by omega))
    _ = _ := by
      simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin]
      rw [nsmul_eq_mul, ←ENNReal.ofReal_natCast,
        ←ENNReal.ofReal_mul (by positivity : 0 ≤ ((T+1:ℕ):ℝ))]
      congr 1
      push_cast
      ring

end

section

universe u
variable {d M T : ℕ} {Ξ : Type u} [MeasurableSpace Ξ]

lemma seededSuccess_eq_integral_pointwise [NeZero d]
    (K : Ξ → FiniteKernelLearner d M T) (ρ : Measure Ξ) [SFinite ρ] (ε : ℝ)
    (hm : JointSuccessMeasurable K ρ ε) :
    seededSuccess K ρ ε = ∫⁻ s, seededPointwiseSuccess K ρ ε s ∂uniformSphere d := by
  let f := fun z : UnitSphere d × (Ξ × Samples d T) =>
    (K z.2.1).conditionalSuccess ε z.1 z.2.2
  let e : ((UnitSphere d × Samples d T) × Ξ) ≃ᵐ UnitSphere d × (Ξ × Samples d T) :=
    MeasurableEquiv.prodAssoc.trans (MeasurableEquiv.prodCongr (MeasurableEquiv.refl _)
      (MeasurableEquiv.prodComm))
  have hp : MeasurePreserving e (((uniformSphere d).prod (sampleLaw d T)).prod ρ)
      ((uniformSphere d).prod (ρ.prod (sampleLaw d T))) :=
    ((MeasurePreserving.id (uniformSphere d)).prod
      (Measure.measurePreserving_swap (μ := sampleLaw d T) (ν := ρ))).comp
      (measurePreserving_prodAssoc (uniformSphere d) (sampleLaw d T) ρ)
  have hf : AEMeasurable f ((uniformSphere d).prod (ρ.prod (sampleLaw d T))) := hm
  have hfD : AEMeasurable (fun z : (UnitSphere d × Samples d T) × Ξ => f (e z))
      (((uniformSphere d).prod (sampleLaw d T)).prod ρ) :=
    hf.comp_quasiMeasurePreserving hp.quasiMeasurePreserving
  calc
    _ = ∫⁻ ξ, ∫⁻ z : UnitSphere d × Samples d T, f (z.1,(ξ,z.2))
          ∂((uniformSphere d).prod (sampleLaw d T)) ∂ρ := rfl
    _ = ∫⁻ z : UnitSphere d × Samples d T, ∫⁻ ξ, f (z.1,(ξ,z.2)) ∂ρ
          ∂((uniformSphere d).prod (sampleLaw d T)) := (lintegral_lintegral_swap hfD).symm
    _ = ∫⁻ z, f (e z) ∂(((uniformSphere d).prod (sampleLaw d T)).prod ρ) :=
      (lintegral_prod _ hfD).symm
    _ = ∫⁻ z, f z ∂((uniformSphere d).prod (ρ.prod (sampleLaw d T))) :=
      hp.lintegral_comp_emb e.measurableEmbedding f
    _ = _ := lintegral_prod _ hf

lemma seededSuccess_of_pointwise [NeZero d]
    (K : Ξ → FiniteKernelLearner d M T) (ρ : Measure Ξ) [SFinite ρ] (ε : ℝ)
    (hm : JointSuccessMeasurable K ρ ε)
    (hs : ∀ s, (2/3 : ℝ≥0∞) ≤ seededPointwiseSuccess K ρ ε s) :
    (2/3 : ℝ≥0∞) ≤ seededSuccess K ρ ε := by
  rw [seededSuccess_eq_integral_pointwise K ρ ε hm]
  calc
    _ = ∫⁻ _ : UnitSphere d, (2/3 : ℝ≥0∞) ∂uniformSphere d := by simp
    _ ≤ _ := lintegral_mono hs

theorem subquadratic_memory_with_pointwise : SubquadraticMemoryWithPointwise.{u} := by
  obtain ⟨c,hc,h⟩ := subquadratic_memory.{u}
  refine ⟨c,hc,?_⟩
  intro M hM
  obtain ⟨d₀,hd₀⟩ := h M hM
  refine ⟨max d₀ 1,?_⟩
  intro d hd T ε hε hε10 Ξ _ ρ hρ K hK hs
  let : IsProbabilityMeasure ρ := hρ
  let : NeZero d := ⟨by omega⟩
  apply hd₀ d (by omega) T ε hε hε10 Ξ ρ hρ K hK
  rcases hs with hs | ⟨hm,hs⟩
  · exact hs
  · exact seededSuccess_of_pointwise K ρ ε hm hs

end

universe u

lemma cap_memory_factor (C : ℝ) {d M : ℕ} (hd : 32 ≤ d)
    (hM : (M:ℝ) ≤ (d:ℝ)^2) :
    Real.exp (C*d)*(2:ℝ)^((M:ℝ)/(d/16:ℕ)) ≤ Real.exp ((C+32)*d) := by
  have hk : 0 < (d/16:ℕ) := by omega
  have hkR : (0:ℝ) < (d/16:ℕ) := by exact_mod_cast hk
  have hdk : d ≤ 32*(d/16) := by omega
  have hdkR : (d:ℝ) ≤ 32*(d/16:ℕ) := by exact_mod_cast hdk
  have hfrac : (M:ℝ)/(d/16:ℕ) ≤ 32*d := by
    rw [div_le_iff₀ hkR]
    nlinarith [Nat.cast_nonneg (α := ℝ) d]
  have hlog : Real.log 2 ≤ 1 := by
    have h := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
    norm_num at h ⊢
    exact h
  rw [Real.rpow_def_of_pos (by norm_num),← Real.exp_add]
  apply Real.exp_le_exp.mpr
  have hnon : (0:ℝ) ≤ (M:ℝ)/(d/16:ℕ) := by positivity
  have hl := mul_le_mul_of_nonneg_right hlog hnon
  nlinarith

lemma cap_finite_to_asymptotic (C : ℝ) {d M : ℕ} (hd : 32 ≤ d)
    (hM : (M:ℝ) ≤ (d:ℝ)^2) (ε α : ℝ) (n : ℕ) (hε : 0 ≤ ε) :
    Real.exp (C*d)*(ε/2)^α*(Real.exp (C*d)*(2:ℝ)^((M:ℝ)/(d/16:ℕ)))^n ≤
      Real.exp ((C+32)*d*(1+n))*(ε/2)^α := by
  have hbase := cap_memory_factor C hd hM
  have hfront : Real.exp (C*d) ≤ Real.exp ((C+32)*d) := by
    apply Real.exp_le_exp.mpr
    nlinarith [Nat.cast_nonneg (α := ℝ) d]
  have ht := mul_le_mul (mul_le_mul_of_nonneg_right hfront (Real.rpow_nonneg (by positivity : 0 ≤ ε/2) α))
    (pow_le_pow_left₀ (by positivity) hbase n) (by positivity) (by positivity)
  apply ht.trans_eq
  rw [← Real.exp_nat_mul]
  calc
    Real.exp ((C+32)*d)*(ε/2)^α*Real.exp ((n:ℝ)*((C+32)*d)) =
        (Real.exp ((C+32)*d)*Real.exp ((n:ℝ)*((C+32)*d)))*(ε/2)^α := by ring
    _ = _ := by rw [← Real.exp_add]; congr 2; ring

theorem cap_fixed_stop_asymptotic :
    ∃ C : ℝ, 0 < C ∧ ∀ M : ℕ → ℕ,
      Asymptotics.IsLittleO atTop (fun d => (M d:ℝ)) (fun d => (d:ℝ)^2) →
      ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d → ∀ T : ℕ, ∀ ε : ℝ,
      0 < ε → ε ≤ 1/10 →
      ∀ (Ξ : Type u) [MeasurableSpace Ξ] (ρ : Measure Ξ), IsProbabilityMeasure ρ →
      ∀ K : Ξ → FiniteKernelLearner d (M d) T, (∀ᵐ ξ ∂ρ, (K ξ).CompletedRules) →
      ∀ t : ℕ, t ≤ T →
      let k := d/16
      let α : ℝ := ((d:ℝ)-1)/2
      seededFixedSuccess K ρ ε t ≤ ENNReal.ofReal
        (Real.exp (C*d*(1+((t+k-1)/k:ℕ)))*(ε/2)^α) := by
  obtain ⟨C,hC,h⟩ := cap_fixed_stop.{u}
  refine ⟨C+32,by linarith,?_⟩
  intro M hM
  obtain ⟨d₀,hd₀⟩ := eventually_atTop.1 hM.eventuallyLE
  refine ⟨max d₀ 32,?_⟩
  intro d hd T ε hε hε10 Ξ _ ρ hρ K hK t ht k α
  have hm : (M d:ℝ) ≤ (d:ℝ)^2 := by
    simpa only [Real.norm_eq_abs,abs_of_nonneg (Nat.cast_nonneg (α := ℝ) (M d)),
      abs_of_nonneg (sq_nonneg (d:ℝ))] using hd₀ d (by omega)
  exact (h d (M d) T (by omega) ε hε hε10 Ξ ρ hρ K hK t ht).trans
    (ENNReal.ofReal_le_ofReal (cap_finite_to_asymptotic C (by omega) hm ε α _ hε.le))

lemma cap_ceil_cost {d T : ℕ} (hd : 32 ≤ d) :
    (d:ℝ)*(1+((T+d/16-1)/(d/16):ℕ)) ≤ 2*d+32*T := by
  let k := d/16
  let n := (T+k-1)/k
  have hk : 0 < k := by dsimp [k]; omega
  have hkR : (0:ℝ) < k := by exact_mod_cast hk
  have hdiv : n*k ≤ T+k := (Nat.div_mul_le_self _ _).trans (by omega)
  have hdivR : (n:ℝ)*k ≤ (T:ℝ)+k := by exact_mod_cast hdiv
  have hdR : (d:ℝ) ≤ 32*k := by exact_mod_cast (show d ≤ 32*k by dsimp [k]; omega)
  have hn : (n:ℝ) ≤ (T:ℝ)/k+1 := by
    have he := div_mul_cancel₀ (T:ℝ) hkR.ne'
    nlinarith
  have hh := mul_le_mul_of_nonneg_left hn (Nat.cast_nonneg (α := ℝ) d)
  have hdk : (d:ℝ)/(k:ℝ) ≤ 32 := (div_le_iff₀ hkR).mpr hdR
  have hT := mul_le_mul_of_nonneg_right hdk (Nat.cast_nonneg (α := ℝ) T)
  change (d:ℝ)*(1+(n:ℝ)) ≤ _
  calc
    _ ≤ (d:ℝ)*((T:ℝ)/k+2) := by nlinarith
    _ = (d:ℝ)/k*T+2*d := by ring
    _ ≤ _ := by linarith
lemma cap_accuracy_cost {d : ℕ} (hd : 3 ≤ d) {ε : ℝ}
    (hε : 0 < ε) (hε1 : ε ≤ 1) :
    (ε/2)^(((d:ℝ)-1)/2) ≤ Real.exp (-(d:ℝ)*Real.log (1/ε)/3) := by
  have hdR : (3:ℝ) ≤ d := by exact_mod_cast hd
  have ha : 0 ≤ ((d:ℝ)-1)/2 := by linarith
  calc
    _ ≤ ε^(((d:ℝ)-1)/2) := Real.rpow_le_rpow (by positivity) (by linarith) ha
    _ ≤ ε^((d:ℝ)/3) := Real.rpow_le_rpow_of_exponent_ge hε hε1 (by linarith)
    _ = _ := by
      rw [Real.rpow_def_of_pos hε,one_div,Real.log_inv]
      congr 1
      ring
theorem cap_total_success_asymptotic :
    ∃ C₁ C₂ : ℝ, 0 < C₁ ∧ 0 < C₂ ∧ ∀ M : ℕ → ℕ,
      Asymptotics.IsLittleO atTop (fun d => (M d:ℝ)) (fun d => (d:ℝ)^2) →
      ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d → ∀ T : ℕ, ∀ ε : ℝ,
      0 < ε → ε ≤ 1/10 →
      ∀ (Ξ : Type u) [MeasurableSpace Ξ] (ρ : Measure Ξ), IsProbabilityMeasure ρ →
      ∀ K : Ξ → FiniteKernelLearner d (M d) T, (∀ᵐ ξ ∂ρ, (K ξ).CompletedRules) →
      seededSuccess K ρ ε ≤ ENNReal.ofReal
        (((T:ℝ)+1)*Real.exp (C₁*d+C₂*T-(d:ℝ)*Real.log (1/ε)/3)) := by
  obtain ⟨C,hC,h⟩ := cap_total_success.{u}
  refine ⟨2*(C+32),32*(C+32),by positivity,by positivity,?_⟩
  intro M hM
  obtain ⟨d₀,hd₀⟩ := eventually_atTop.1 hM.eventuallyLE
  refine ⟨max d₀ 32,?_⟩
  intro d hd T ε hε hε10 Ξ _ ρ hρ K hK
  have hm : (M d:ℝ) ≤ (d:ℝ)^2 := by
    simpa only [Real.norm_eq_abs,abs_of_nonneg (Nat.cast_nonneg (α := ℝ) (M d)),
      abs_of_nonneg (sq_nonneg (d:ℝ))] using hd₀ d (by omega)
  apply (h d (M d) T (by omega) ε hε hε10 Ξ ρ hρ K hK).trans
  apply ENNReal.ofReal_le_ofReal
  simp only [mul_assoc]
  apply mul_le_mul_of_nonneg_left _ (by positivity : (0:ℝ) ≤ (T:ℝ)+1)
  have hfinite := cap_finite_to_asymptotic C (by omega) hm ε (((d:ℝ)-1)/2)
    ((T+d/16-1)/(d/16)) hε.le
  simp only [mul_assoc] at hfinite
  apply hfinite.trans
  have hcost := mul_le_mul_of_nonneg_left (cap_ceil_cost (T := T) (by omega : 32 ≤ d))
    (by positivity : (0:ℝ) ≤ C+32)
  have he : Real.exp ((C+32)*d*(1+((T+d/16-1)/(d/16):ℕ))) ≤
      Real.exp (2*(C+32)*d+32*(C+32)*T) := by
    apply Real.exp_le_exp.mpr
    nlinarith
  simp only [mul_assoc] at he
  apply (mul_le_mul he (cap_accuracy_cost (d := d) (by omega) hε (by linarith)) (by positivity) (by positivity)).trans_eq
  rw [← Real.exp_add]
  congr 1
  ring
end FiniteKernelLearner
end NoiselessRegression
end
end OAI

	import OAI.Probability.ProjectionMoments.Seeded

	namespace OAI

	noncomputable section

	open MeasureTheory ProbabilityTheory Set Filter
	open scoped ENNReal NNReal RealInnerProductSpace
	namespace NoiselessRegression
	namespace Learner
	variable {d M T : ℕ} {Ω : Type*} [MeasurableSpace Ω]

	lemma uniformSuccess_le_sum_fixed (L : Learner d M T Ω) (ρ : Measure Ω) (ε : ℝ) :
	L.uniformSuccess ρ ε ≤ ∑ t : Fin (T+1), L.uniformFixedSuccess ρ ε t := by
	classical
	let μ := (uniformSphere d).prod (ρ.prod (sampleLaw d T))
	have hs : {z : UnitSphere d × (Ω × Samples d T) \|
	angularError z.1 (L.estimate z.1 z.2.1 z.2.2) ≤ ε} =
	⋃ t : Fin (T+1), {z \| (L.terminal z.1 z.2.1 z.2.2).1 = t ∧
	angularError z.1 (L.estimate z.1 z.2.1 z.2.2) ≤ ε} := by
	ext z
	simp only [Set.mem_ofPred_eq,Set.mem_iUnion]
	constructor
	· intro hz
	obtain ⟨t,ht,he⟩ := L.terminal_forced_witness z.1 z.2.1 z.2.2
	exact ⟨⟨t,by omega⟩,by simpa [he] using hz⟩
	· rintro ⟨t,ht,hz⟩
	exact hz
	change μ _ ≤ ∑ t : Fin (T+1), μ _
	rw [hs]
	exact (measure_iUnion_le _).trans_eq (tsum_fintype _)

	end Learner
	namespace FiniteKernelLearner

	section
	universe u

	theorem cap_total_success :
	∃ C : ℝ, 0 < C ∧ ∀ d M T : ℕ, 32 ≤ d → ∀ ε : ℝ,
	0 < ε → ε ≤ 1/10 →
	∀ (Ξ : Type u) [MeasurableSpace Ξ] (ρ : Measure Ξ), IsProbabilityMeasure ρ →
	∀ K : Ξ → FiniteKernelLearner d M T, (∀ᵐ ξ ∂ρ, (K ξ).CompletedRules) →
	let k := d/16
	let α : ℝ := ((d:ℝ)-1)/2
	seededSuccess K ρ ε ≤ ENNReal.ofReal
	(((T:ℝ)+1)Real.exp (Cd)(ε/2)^α
	(Real.exp (Cd)(2:ℝ)^((M:ℝ)/k))^((T+k-1)/k)) := by
	obtain ⟨C,hC,h⟩ := Learner.cap_fixed_stop.{0}
	refine ⟨C,hC,?_⟩
	intro d M T hd ε hε hε10 Ξ _ ρ hρ K hK k α
	let : IsProbabilityMeasure ρ := hρ
	apply seed_average_bound
	filter_upwards [hK] with ξ hξ
	obtain ⟨ν,L,hν,hB,hL,he,_⟩ := (K ξ).exists_equivalent_tape hξ (by omega)
	rw [←he ε]
	have hs := L.uniformSuccess_le_sum_fixed ν ε
	have hbase : 1 ≤ Real.exp (Cd)(2:ℝ)^((M:ℝ)/k) :=
	one_le_mul_of_one_le_of_one_le (Real.one_le_exp (by positivity))
	(Real.one_le_rpow (by norm_num) (by positivity))
	apply hs.trans
	calc
	_ ≤ ∑ _t : Fin (T+1), ENNReal.ofReal
	(Real.exp (Cd)(ε/2)^α(Real.exp (Cd)*(2:ℝ)^((M:ℝ)/k))^((T+k-1)/k)) := by
	apply Finset.sum_le_sum
	intro t _
	apply (h d M T hd ε hε hε10 (Tape M T) ν hν L hL
	(L.measurable_terminal hB).fst.aemeasurable t (by omega)).trans
	apply ENNReal.ofReal_le_ofReal
	apply mul_le_mul_of_nonneg_left _ (by positivity)
	exact pow_le_pow_right₀ hbase (Nat.div_le_div_right (by omega))
	_ = _ := by
	simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin]
	rw [nsmul_eq_mul, ←ENNReal.ofReal_natCast,
	←ENNReal.ofReal_mul (by positivity : 0 ≤ ((T+1:ℕ):ℝ))]
	congr 1
	push_cast
	ring

	end

	section

	universe u
	variable {d M T : ℕ} {Ξ : Type u} [MeasurableSpace Ξ]

	lemma seededSuccess_eq_integral_pointwise [NeZero d]
	(K : Ξ → FiniteKernelLearner d M T) (ρ : Measure Ξ) [SFinite ρ] (ε : ℝ)
	(hm : JointSuccessMeasurable K ρ ε) :
	seededSuccess K ρ ε = ∫⁻ s, seededPointwiseSuccess K ρ ε s ∂uniformSphere d := by
	let f := fun z : UnitSphere d × (Ξ × Samples d T) =>
	(K z.2.1).conditionalSuccess ε z.1 z.2.2
	let e : ((UnitSphere d × Samples d T) × Ξ) ≃ᵐ UnitSphere d × (Ξ × Samples d T) :=
	MeasurableEquiv.prodAssoc.trans (MeasurableEquiv.prodCongr (MeasurableEquiv.refl _)
	(MeasurableEquiv.prodComm))
	have hp : MeasurePreserving e (((uniformSphere d).prod (sampleLaw d T)).prod ρ)
	((uniformSphere d).prod (ρ.prod (sampleLaw d T))) :=
	((MeasurePreserving.id (uniformSphere d)).prod
	(Measure.measurePreserving_swap (μ := sampleLaw d T) (ν := ρ))).comp
	(measurePreserving_prodAssoc (uniformSphere d) (sampleLaw d T) ρ)
	have hf : AEMeasurable f ((uniformSphere d).prod (ρ.prod (sampleLaw d T))) := hm
	have hfD : AEMeasurable (fun z : (UnitSphere d × Samples d T) × Ξ => f (e z))
	(((uniformSphere d).prod (sampleLaw d T)).prod ρ) :=
	hf.comp_quasiMeasurePreserving hp.quasiMeasurePreserving
	calc
	_ = ∫⁻ ξ, ∫⁻ z : UnitSphere d × Samples d T, f (z.1,(ξ,z.2))
	∂((uniformSphere d).prod (sampleLaw d T)) ∂ρ := rfl
	_ = ∫⁻ z : UnitSphere d × Samples d T, ∫⁻ ξ, f (z.1,(ξ,z.2)) ∂ρ
	∂((uniformSphere d).prod (sampleLaw d T)) := (lintegral_lintegral_swap hfD).symm
	_ = ∫⁻ z, f (e z) ∂(((uniformSphere d).prod (sampleLaw d T)).prod ρ) :=
	(lintegral_prod _ hfD).symm
	_ = ∫⁻ z, f z ∂((uniformSphere d).prod (ρ.prod (sampleLaw d T))) :=
	hp.lintegral_comp_emb e.measurableEmbedding f
	_ = _ := lintegral_prod _ hf

	lemma seededSuccess_of_pointwise [NeZero d]
	(K : Ξ → FiniteKernelLearner d M T) (ρ : Measure Ξ) [SFinite ρ] (ε : ℝ)
	(hm : JointSuccessMeasurable K ρ ε)
	(hs : ∀ s, (2/3 : ℝ≥0∞) ≤ seededPointwiseSuccess K ρ ε s) :
	(2/3 : ℝ≥0∞) ≤ seededSuccess K ρ ε := by
	rw [seededSuccess_eq_integral_pointwise K ρ ε hm]
	calc
	_ = ∫⁻ _ : UnitSphere d, (2/3 : ℝ≥0∞) ∂uniformSphere d := by simp
	_ ≤ _ := lintegral_mono hs

	theorem subquadratic_memory_with_pointwise : SubquadraticMemoryWithPointwise.{u} := by
	obtain ⟨c,hc,h⟩ := subquadratic_memory.{u}
	refine ⟨c,hc,?_⟩
	intro M hM
	obtain ⟨d₀,hd₀⟩ := h M hM
	refine ⟨max d₀ 1,?_⟩
	intro d hd T ε hε hε10 Ξ _ ρ hρ K hK hs
	let : IsProbabilityMeasure ρ := hρ
	let : NeZero d := ⟨by omega⟩
	apply hd₀ d (by omega) T ε hε hε10 Ξ ρ hρ K hK
	rcases hs with hs \| ⟨hm,hs⟩
	· exact hs
	· exact seededSuccess_of_pointwise K ρ ε hm hs

	end

	universe u

	lemma cap_memory_factor (C : ℝ) {d M : ℕ} (hd : 32 ≤ d)
	(hM : (M:ℝ) ≤ (d:ℝ)^2) :
	Real.exp (Cd)(2:ℝ)^((M:ℝ)/(d/16:ℕ)) ≤ Real.exp ((C+32)*d) := by
	have hk : 0 < (d/16:ℕ) := by omega
	have hkR : (0:ℝ) < (d/16:ℕ) := by exact_mod_cast hk
	have hdk : d ≤ 32*(d/16) := by omega
	have hdkR : (d:ℝ) ≤ 32*(d/16:ℕ) := by exact_mod_cast hdk
	have hfrac : (M:ℝ)/(d/16:ℕ) ≤ 32*d := by
	rw [div_le_iff₀ hkR]
	nlinarith [Nat.cast_nonneg (α := ℝ) d]
	have hlog : Real.log 2 ≤ 1 := by
	have h := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
	norm_num at h ⊢
	exact h
	rw [Real.rpow_def_of_pos (by norm_num),← Real.exp_add]
	apply Real.exp_le_exp.mpr
	have hnon : (0:ℝ) ≤ (M:ℝ)/(d/16:ℕ) := by positivity
	have hl := mul_le_mul_of_nonneg_right hlog hnon
	nlinarith

	lemma cap_finite_to_asymptotic (C : ℝ) {d M : ℕ} (hd : 32 ≤ d)
	(hM : (M:ℝ) ≤ (d:ℝ)^2) (ε α : ℝ) (n : ℕ) (hε : 0 ≤ ε) :
	Real.exp (Cd)(ε/2)^α(Real.exp (Cd)*(2:ℝ)^((M:ℝ)/(d/16:ℕ)))^n ≤
	Real.exp ((C+32)d(1+n))*(ε/2)^α := by
	have hbase := cap_memory_factor C hd hM
	have hfront : Real.exp (Cd) ≤ Real.exp ((C+32)d) := by
	apply Real.exp_le_exp.mpr
	nlinarith [Nat.cast_nonneg (α := ℝ) d]
	have ht := mul_le_mul (mul_le_mul_of_nonneg_right hfront (Real.rpow_nonneg (by positivity : 0 ≤ ε/2) α))
	(pow_le_pow_left₀ (by positivity) hbase n) (by positivity) (by positivity)
	apply ht.trans_eq
	rw [← Real.exp_nat_mul]
	calc
	Real.exp ((C+32)d)(ε/2)^αReal.exp ((n:ℝ)((C+32)*d)) =
	(Real.exp ((C+32)d)Real.exp ((n:ℝ)((C+32)d)))*(ε/2)^α := by ring
	_ = _ := by rw [← Real.exp_add]; congr 2; ring

	theorem cap_fixed_stop_asymptotic :
	∃ C : ℝ, 0 < C ∧ ∀ M : ℕ → ℕ,
	Asymptotics.IsLittleO atTop (fun d => (M d:ℝ)) (fun d => (d:ℝ)^2) →
	∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d → ∀ T : ℕ, ∀ ε : ℝ,
	0 < ε → ε ≤ 1/10 →
	∀ (Ξ : Type u) [MeasurableSpace Ξ] (ρ : Measure Ξ), IsProbabilityMeasure ρ →
	∀ K : Ξ → FiniteKernelLearner d (M d) T, (∀ᵐ ξ ∂ρ, (K ξ).CompletedRules) →
	∀ t : ℕ, t ≤ T →
	let k := d/16
	let α : ℝ := ((d:ℝ)-1)/2
	seededFixedSuccess K ρ ε t ≤ ENNReal.ofReal
	(Real.exp (Cd(1+((t+k-1)/k:ℕ)))*(ε/2)^α) := by
	obtain ⟨C,hC,h⟩ := cap_fixed_stop.{u}
	refine ⟨C+32,by linarith,?_⟩
	intro M hM
	obtain ⟨d₀,hd₀⟩ := eventually_atTop.1 hM.eventuallyLE
	refine ⟨max d₀ 32,?_⟩
	intro d hd T ε hε hε10 Ξ _ ρ hρ K hK t ht k α
	have hm : (M d:ℝ) ≤ (d:ℝ)^2 := by
	simpa only [Real.norm_eq_abs,abs_of_nonneg (Nat.cast_nonneg (α := ℝ) (M d)),
	abs_of_nonneg (sq_nonneg (d:ℝ))] using hd₀ d (by omega)
	exact (h d (M d) T (by omega) ε hε hε10 Ξ ρ hρ K hK t ht).trans
	(ENNReal.ofReal_le_ofReal (cap_finite_to_asymptotic C (by omega) hm ε α _ hε.le))

	lemma cap_ceil_cost {d T : ℕ} (hd : 32 ≤ d) :
	(d:ℝ)(1+((T+d/16-1)/(d/16):ℕ)) ≤ 2d+32*T := by
	let k := d/16
	let n := (T+k-1)/k
	have hk : 0 < k := by dsimp [k]; omega
	have hkR : (0:ℝ) < k := by exact_mod_cast hk
	have hdiv : n*k ≤ T+k := (Nat.div_mul_le_self _ _).trans (by omega)
	have hdivR : (n:ℝ)*k ≤ (T:ℝ)+k := by exact_mod_cast hdiv
	have hdR : (d:ℝ) ≤ 32k := by exact_mod_cast (show d ≤ 32k by dsimp [k]; omega)
	have hn : (n:ℝ) ≤ (T:ℝ)/k+1 := by
	have he := div_mul_cancel₀ (T:ℝ) hkR.ne'
	nlinarith
	have hh := mul_le_mul_of_nonneg_left hn (Nat.cast_nonneg (α := ℝ) d)
	have hdk : (d:ℝ)/(k:ℝ) ≤ 32 := (div_le_iff₀ hkR).mpr hdR
	have hT := mul_le_mul_of_nonneg_right hdk (Nat.cast_nonneg (α := ℝ) T)
	change (d:ℝ)*(1+(n:ℝ)) ≤ _
	calc
	_ ≤ (d:ℝ)*((T:ℝ)/k+2) := by nlinarith
	_ = (d:ℝ)/kT+2d := by ring
	_ ≤ _ := by linarith

	lemma cap_accuracy_cost {d : ℕ} (hd : 3 ≤ d) {ε : ℝ}
	(hε : 0 < ε) (hε1 : ε ≤ 1) :
	(ε/2)^(((d:ℝ)-1)/2) ≤ Real.exp (-(d:ℝ)*Real.log (1/ε)/3) := by
	have hdR : (3:ℝ) ≤ d := by exact_mod_cast hd
	have ha : 0 ≤ ((d:ℝ)-1)/2 := by linarith
	calc
	_ ≤ ε^(((d:ℝ)-1)/2) := Real.rpow_le_rpow (by positivity) (by linarith) ha
	_ ≤ ε^((d:ℝ)/3) := Real.rpow_le_rpow_of_exponent_ge hε hε1 (by linarith)
	_ = _ := by
	rw [Real.rpow_def_of_pos hε,one_div,Real.log_inv]
	congr 1
	ring

	theorem cap_total_success_asymptotic :
	∃ C₁ C₂ : ℝ, 0 < C₁ ∧ 0 < C₂ ∧ ∀ M : ℕ → ℕ,
	Asymptotics.IsLittleO atTop (fun d => (M d:ℝ)) (fun d => (d:ℝ)^2) →
	∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d → ∀ T : ℕ, ∀ ε : ℝ,
	0 < ε → ε ≤ 1/10 →
	∀ (Ξ : Type u) [MeasurableSpace Ξ] (ρ : Measure Ξ), IsProbabilityMeasure ρ →
	∀ K : Ξ → FiniteKernelLearner d (M d) T, (∀ᵐ ξ ∂ρ, (K ξ).CompletedRules) →
	seededSuccess K ρ ε ≤ ENNReal.ofReal
	(((T:ℝ)+1)Real.exp (C₁d+C₂T-(d:ℝ)Real.log (1/ε)/3)) := by
	obtain ⟨C,hC,h⟩ := cap_total_success.{u}
	refine ⟨2(C+32),32(C+32),by positivity,by positivity,?_⟩
	intro M hM
	obtain ⟨d₀,hd₀⟩ := eventually_atTop.1 hM.eventuallyLE
	refine ⟨max d₀ 32,?_⟩
	intro d hd T ε hε hε10 Ξ _ ρ hρ K hK
	have hm : (M d:ℝ) ≤ (d:ℝ)^2 := by
	simpa only [Real.norm_eq_abs,abs_of_nonneg (Nat.cast_nonneg (α := ℝ) (M d)),
	abs_of_nonneg (sq_nonneg (d:ℝ))] using hd₀ d (by omega)
	apply (h d (M d) T (by omega) ε hε hε10 Ξ ρ hρ K hK).trans
	apply ENNReal.ofReal_le_ofReal
	simp only [mul_assoc]
	apply mul_le_mul_of_nonneg_left _ (by positivity : (0:ℝ) ≤ (T:ℝ)+1)
	have hfinite := cap_finite_to_asymptotic C (by omega) hm ε (((d:ℝ)-1)/2)
	((T+d/16-1)/(d/16)) hε.le
	simp only [mul_assoc] at hfinite
	apply hfinite.trans
	have hcost := mul_le_mul_of_nonneg_left (cap_ceil_cost (T := T) (by omega : 32 ≤ d))
	(by positivity : (0:ℝ) ≤ C+32)
	have he : Real.exp ((C+32)d(1+((T+d/16-1)/(d/16):ℕ))) ≤
	Real.exp (2(C+32)d+32(C+32)T) := by
	apply Real.exp_le_exp.mpr
	nlinarith
	simp only [mul_assoc] at he
	apply (mul_le_mul he (cap_accuracy_cost (d := d) (by omega) hε (by linarith)) (by positivity) (by positivity)).trans_eq
	rw [← Real.exp_add]
	congr 1
	ring

	end FiniteKernelLearner
	end NoiselessRegression

	end

	end OAI