| import Mathlib |
| import OAI.Probability.Perceptron.Variational.BackwardScore |
| import OAI.Probability.Perceptron.Variational.CoupledDual |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal ENNReal BigOperators BoundedContinuousFunction |
| namespace SphericalPerceptronFreeEnergy |
|
|
| lemma Jet3.eq_of_f_eq {g h : Jet3} (he : g.f=h.f) : g=h := by |
| have h1 : g.d1=h.d1 := by |
| ext x |
| rw [← g.deriv_eq,← h.deriv_eq,he] |
| have h2 : g.d2=h.d2 := by |
| ext x |
| rw [← g.deriv2_eq,← h.deriv2_eq,he] |
| have h3 : g.d3=h.d3 := by |
| ext x |
| rw [← g.deriv3_eq,← h.deriv3_eq,he] |
| cases g |
| cases h |
| simp_all |
|
|
| lemma Jet3.heatLog_zero_time (g : Jet3) (d : ℝ≥0) : g.heatLog 0 d=g := by |
| apply Jet3.eq_of_f_eq |
| ext x |
| simp only [Jet3.heatLog_f,heatLogBCF_coe,heatLog_zero] |
|
|
| def finiteStepControlJet (g : Jet3) : (n : ℕ)→(Fin (n+1)→Time)→(Fin n→ℝ≥0)→Jet3 |
| | 0,_,_ => g |
| | n+1,q,d => (finiteStepControlJet g n (fun i => q i.succ) (fun i => d i.succ)).heatLog |
| (timeSpan (q 0) (q 1)) (d 0) |
|
|
| lemma finiteStepControlJet_chain (g : Jet3) (m : Trial) (n : ℕ) |
| (q : Fin (n+1)→Time) (d : Fin n→ℝ≥0) (hq : Monotone q) |
| (hm : ∀ i : Fin n, ∀ r∈Ioc (q i.castSucc) (q i.succ), m r=(d i:ℝ)) : |
| HeatChain m g (q 0) (q (Fin.last n)) (finiteStepControlJet g n q d) := by |
| induction n with |
| | zero => exact HeatChain.nil (q 0) |
| | succ n ih => |
| have ih' := ih (fun i => q i.succ) (fun i => d i.succ) |
| (hq.comp (Fin.strictMono_succ.monotone)) (fun i => hm i.succ) |
| change HeatChain m g (q 1) (q (Fin.last (n+1))) _ at ih' |
| have h01 : q 0≤q 1 := hq (Fin.zero_le _) |
| rcases lt_or_eq_of_le h01 with hlt | he |
| · exact HeatChain.cons (d 0) hlt (hq (Fin.le_last _)) (hm 0) ih' |
| · simpa only [finiteStepControlJet,he,timeSpan,sub_self,Real.toNNReal_zero, |
| Jet3.heatLog_zero_time] using ih' |
|
|
| lemma finiteStepControlJet_value (g : Jet3) (m : Trial) (n : ℕ) |
| (q : Fin (n+1)→Time) (d : Fin n→ℝ≥0) (hq : Monotone q) |
| (hq0 : q 0=0) (hq1 : q (Fin.last n)=1) |
| (hm : ∀ i : Fin n, ∀ r∈Ioc (q i.castSucc) (q i.succ), m r=(d i:ℝ)) |
| (P : Measure BrownianPath) [IsProbabilityMeasure P] (hB : IsBrownianReal brownianEval P) : |
| controlValue P g.f m=(finiteStepControlJet g n q d).f 0 := by |
| have H := finiteStepControlJet_chain g m n q d hq hm |
| rw [hq0,hq1] at H |
| exact H.controlValue_eq P hB |
|
|
| def leftWeightedStepTrial {I : Type*} [Fintype I] (w : I→ℝ) (q : I→Time) |
| (hw : ∀ i, 0≤w i) (hw1 : ∑ i, w i=1) : Trial where |
| toFun := fun t => ∑ i, if q i<t then w i else 0 |
| monotone := by |
| intro s t hst |
| apply Finset.sum_le_sum |
| intro i _ |
| by_cases hi : q i<s |
| · simp only [ite_eq_left hi,ite_eq_left (hi.trans_le hst),le_refl] |
| · rw [ite_eq_right hi] |
| split_ifs <;> first | exact hw i | rfl |
| measurable := by |
| apply Finset.measurable_sum |
| intro i _ |
| exact measurable_const.ite (measurableSet_lt measurable_const measurable_id) measurable_const |
| nonneg := by |
| intro t |
| apply Finset.sum_nonneg |
| intro i _ |
| split_ifs <;> first | exact hw i | rfl |
| le_one := by |
| intro t |
| rw [← hw1] |
| apply Finset.sum_le_sum |
| intro i _ |
| split_ifs <;> first | rfl | exact hw i |
| |
| lemma leftWeightedStepTrial_ae {I : Type*} [Fintype I] (w : I→ℝ) (q : I→Time) |
| (hw : ∀ i, 0≤w i) (hw1 : ∑ i, w i=1) : |
| (leftWeightedStepTrial w q hw hw1 : Time→ℝ)=ᵐ[timeLaw] weightedStepTrial w q hw hw1 := by |
| have he : ∀ᵐ t ∂timeLaw, ∀ i, t≠q i := by |
| apply ae_all_iff.mpr |
| intro i |
| apply ae_iff.mpr |
| simp |
| filter_upwards [he] with t ht |
| apply Finset.sum_congr rfl |
| intro i _ |
| by_cases hi : q i≤t |
| · rw [ite_eq_left hi,ite_eq_left (lt_of_le_of_ne hi (Ne.symm (ht i)))] |
| · rw [ite_eq_right hi,ite_eq_right (not_lt_of_ge (le_of_not_ge hi))] |
|
|
| def weightedControlCoefficient (n : ℕ) (w : Fin (n+1)→ℝ) (hw : ∀ i, 0≤w i) |
| (i : Fin n) : ℝ≥0 := |
| ⟨∑ j : Fin (n+1), if j ≤ i.castSucc then w j else 0,Finset.sum_nonneg fun j _ => by |
| split_ifs <;> first | exact hw j | rfl⟩ |
| |
| lemma leftWeightedStepTrial_interval (n : ℕ) (w : Fin (n+1)→ℝ) (q : Fin (n+1)→Time) |
| (hw : ∀ i, 0≤w i) (hw1 : ∑ i, w i=1) (hq : Monotone q) |
| (i : Fin n) {r : Time} (hr : r∈Ioc (q i.castSucc) (q i.succ)) : |
| leftWeightedStepTrial w q hw hw1 r=(weightedControlCoefficient n w hw i:ℝ) := by |
| change (∑ j, if q j<r then w j else 0) = ∑ j, if j ≤ i.castSucc then w j else 0 |
| apply Finset.sum_congr rfl |
| intro j _ |
| have he : q j<r ↔ j ≤ i.castSucc := by |
| constructor |
| · intro hj |
| by_contra hn |
| have hij : i.succ ≤ j := by |
| simp only [Fin.le_iff_val_le_val,Fin.val_succ,Fin.val_castSucc] at hn ⊢ |
| omega |
| exact not_lt_of_ge ((hq hij).trans' hr.2) hj |
| · intro hj |
| exact (hq hj).trans_lt hr.1 |
| simp only [he] |
|
|
| theorem weightedStepTrial_controlValue (n : ℕ) (w : Fin (n+1)→ℝ) (q : Fin (n+1)→Time) |
| (hw : ∀ i, 0≤w i) (hw1 : ∑ i, w i=1) (hq : Monotone q) |
| (hq0 : q 0=0) (hq1 : q (Fin.last n)=1) (g : Jet3) |
| (P : Measure BrownianPath) [IsProbabilityMeasure P] (hB : IsBrownianReal brownianEval P) : |
| controlValue P g.f (weightedStepTrial w q hw hw1)= |
| (finiteStepControlJet g n q (weightedControlCoefficient n w hw)).f 0 := by |
| have he := leftWeightedStepTrial_ae w q hw hw1 |
| have hv := controlValue_trial_abs_sub_le P g.f (weightedStepTrial w q hw hw1) |
| (leftWeightedStepTrial w q hw hw1) ‖g.d1‖₊ g.lipschitz |
| have hz : (∫ t, |weightedStepTrial w q hw hw1 t-leftWeightedStepTrial w q hw hw1 t| ∂timeLaw)=0 := by |
| apply integral_eq_zero_of_ae |
| filter_upwards [he] with t ht |
| simp only [ht,sub_self,abs_zero,Pi.zero_apply] |
| rw [hz,mul_zero] at hv |
| have hv' := sub_eq_zero.mp (abs_eq_zero.mp (le_antisymm hv (abs_nonneg _))) |
| rw [hv'] |
| exact finiteStepControlJet_value g _ n q (weightedControlCoefficient n w hw) hq hq0 hq1 |
| (fun i r hr => leftWeightedStepTrial_interval n w q hw hw1 hq i hr) P hB |
|
|
| def paddedControlNodes {n : ℕ} (q : Fin (n+1)→Time) : Fin (n+3)→Time := |
| Fin.cons 0 (Fin.snoc q 1) |
|
|
| def paddedControlWeights {n : ℕ} (w : Fin (n+1)→ℝ) : Fin (n+3)→ℝ := |
| Fin.cons 0 (Fin.snoc w 0) |
|
|
| lemma paddedControlWeights_nonneg {n : ℕ} (w : Fin (n+1)→ℝ) (hw : ∀ i, 0≤w i) : |
| ∀ i, 0≤paddedControlWeights w i := by |
| intro i |
| refine Fin.cases (by simp [paddedControlWeights]) (fun j => ?_) i |
| change 0≤(Fin.snoc w 0 : Fin (n+2)→ℝ) j |
| refine Fin.lastCases (by simp) (fun l => ?_) j |
| simpa using hw l |
|
|
| lemma paddedControlWeights_sum {n : ℕ} (w : Fin (n+1)→ℝ) (hw1 : ∑ i, w i=1) : |
| ∑ i, paddedControlWeights w i=1 := by |
| unfold paddedControlWeights |
| rw [Fin.sum_univ_succ] |
| simp only [Fin.cons_zero,Fin.cons_succ,zero_add] |
| rw [Fin.sum_univ_castSucc] |
| simpa using hw1 |
|
|
| lemma paddedControlNodes_mono {n : ℕ} (q : Fin (n+1)→Time) (hq : Monotone q) : |
| Monotone (paddedControlNodes q) := by |
| have hs : Monotone (Fin.snoc q 1) := by |
| simpa only [Fin.insertNth_last'] using Fin.insertNth_last_monotone hq 1 (show q (Fin.last n)≤1 from le_top) |
| simpa only [Fin.insertNth_zero',paddedControlNodes] using |
| Fin.insertNth_zero_monotone hs 0 (show (0:Time)≤(Fin.snoc q 1 : Fin (n+2)→Time) 0 from bot_le) |
|
|
| lemma paddedControlTrial_eq {n : ℕ} (w : Fin (n+1)→ℝ) (q : Fin (n+1)→Time) |
| (hw : ∀ i, 0≤w i) (hw1 : ∑ i, w i=1) : |
| weightedStepTrial (paddedControlWeights w) (paddedControlNodes q) |
| (paddedControlWeights_nonneg w hw) (paddedControlWeights_sum w hw1)= |
| weightedStepTrial w q hw hw1 := by |
| apply Trial.ext_fun |
| intro t |
| change (∑ i, if paddedControlNodes q i≤t then paddedControlWeights w i else 0)= |
| ∑ i, if q i≤t then w i else 0 |
| unfold paddedControlNodes paddedControlWeights |
| rw [Fin.sum_univ_succ] |
| simp only [Fin.cons_zero,Fin.cons_succ,ite_self,zero_add] |
| rw [Fin.sum_univ_castSucc] |
| simp only [Fin.snoc_castSucc,Fin.snoc_last,ite_self,add_zero] |
|
|
| theorem sourceWeightedTrial_controlValue (n : ℕ) (w : Fin (n+1)→ℝ) (q : Fin (n+1)→Time) |
| (hw : ∀ i, 0≤w i) (hw1 : ∑ i, w i=1) (hq : Monotone q) (g : Jet3) |
| (P : Measure BrownianPath) [IsProbabilityMeasure P] (hB : IsBrownianReal brownianEval P) : |
| controlValue P g.f (weightedStepTrial w q hw hw1)= |
| (finiteStepControlJet g (n+2) (paddedControlNodes q) |
| (weightedControlCoefficient (n+2) (paddedControlWeights w) (paddedControlWeights_nonneg w hw))).f 0 := by |
| rw [← paddedControlTrial_eq w q hw hw1] |
| exact weightedStepTrial_controlValue (n+2) _ _ (paddedControlWeights_nonneg w hw) |
| (paddedControlWeights_sum w hw1) (paddedControlNodes_mono q hq) |
| (by simp [paddedControlNodes]) (by simp [paddedControlNodes]) g P hB |
|
|
| end SphericalPerceptronFreeEnergy |
| end |
|
|
| end OAI |
|
|
