| import Mathlib | |
| import OAI.Probability.Perceptron.Variational.DepthBlock | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set Filter | |
| open scoped Classical ENNReal NNReal BigOperators Topology | |
| namespace SphericalPerceptronFreeEnergy | |
| lemma indexedDepthLaw_weight_tail (k : ℕ) (w : Fin (k+1) → ℝ) | |
| (hw : ∀ i, 0 < w i) (hw1 : ∑ i, w i = 1) (d : ℕ) : | |
| (indexedDepthLaw k (stepCumulative w) : Measure (FiniteOverlap (Fin (k+1)))).real | |
| {R | d ≤ (R 0 1).val} = ∑ i, if d ≤ i.val then w i else 0 := by | |
| by_cases hdk : d ≤ k | |
| · have hh := congrArg ENNReal.toReal (indexedDepthLaw_pair_tail k (stepCumulative w) | |
| (stepCumulative_strictMono w hw) (stepCumulative_pos w hw) | |
| (stepCumulative_lt_one w hw hw1) d hdk) | |
| have hb : (if h : d = 0 then (0:ℝ) else stepCumulative w ⟨d-1,by omega⟩) ≤ 1 := by | |
| split_ifs | |
| · norm_num | |
| · exact (stepCumulative_lt_one w hw hw1 _).le | |
| have he : (indexedDepthLaw k (stepCumulative w) : Measure (FiniteOverlap (Fin (k+1)))).real | |
| {R | d ≤ (R 0 1).val} = | |
| 1 - (if h : d = 0 then 0 else stepCumulative w ⟨d-1,by omega⟩) := by | |
| simpa only [measureReal_def,ENNReal.toReal_ofReal (sub_nonneg.mpr hb)] using hh | |
| rw [he] | |
| by_cases hd0 : d = 0 | |
| · subst d | |
| simpa only [dite_true,Nat.zero_le,ite_true,sub_zero] using hw1.symm | |
| · rw [dite_eq_right hd0,←hw1] | |
| unfold stepCumulative | |
| rw [←Finset.sum_sub_distrib] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| have hiff : i ≤ (⟨d-1,by omega⟩ : Fin k).castSucc ↔ ¬ d ≤ i.val := by | |
| change i.val ≤ d-1 ↔ ¬ d ≤ i.val | |
| omega | |
| by_cases hi : d ≤ i.val | |
| · simp only [ite_eq_left hi,ite_eq_right (fun h => (hiff.mp h) hi),sub_zero] | |
| · simp only [ite_eq_right hi,ite_eq_left (hiff.mpr hi),sub_self] | |
| · have hf (l : Fin (k+1)) : ¬ d ≤ l.val := by omega | |
| have he : {R : FiniteOverlap (Fin (k+1)) | d ≤ (R 0 1).val} = ∅ := by | |
| ext R | |
| simp only [Set.mem_ofPred_eq,Set.mem_empty_iff_false,iff_false] | |
| exact hf _ | |
| rw [he] | |
| simp only [measureReal_empty,hf,ite_false,Finset.sum_const_zero] | |
| theorem indexedDepthLaw_pair_weights (k : ℕ) (w : Fin (k+1) → ℝ) | |
| (hw : ∀ i, 0 < w i) (hw1 : ∑ i, w i = 1) (c : Fin (k+1)) : | |
| (indexedDepthLaw k (stepCumulative w) : Measure (FiniteOverlap (Fin (k+1)))).real | |
| {R | R 0 1 = c} = w c := by | |
| have h := measureReal_fin_atom_from_tails | |
| (indexedDepthLaw k (stepCumulative w) : Measure (FiniteOverlap (Fin (k+1)))) | |
| k (fun R => R 0 1) (by fun_prop) univ MeasurableSet.univ c | |
| simp only [Set.mem_univ,true_and] at h | |
| rw [h,indexedDepthLaw_weight_tail k w hw hw1,indexedDepthLaw_weight_tail k w hw hw1, | |
| ←Finset.sum_sub_distrib] | |
| calc | |
| _ = ∑ i, if i = c then w c else 0 := by | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| rcases lt_trichotomy i c with hi | rfl | hi | |
| · have h1 : ¬ c.val ≤ i.val := by exact not_le.mpr hi | |
| have h2 : ¬ c.val+1 ≤ i.val := by omega | |
| simp only [ite_eq_right h1,ite_eq_right h2,ite_eq_right (ne_of_lt hi),sub_self] | |
| · simp | |
| · have h1 : c.val ≤ i.val := le_of_lt hi | |
| have h2 : c.val+1 ≤ i.val := hi | |
| simp only [ite_eq_left h1,ite_eq_left h2,ite_eq_right (ne_of_gt hi),sub_self] | |
| _ = w c := by simp | |
| end SphericalPerceptronFreeEnergy | |
| end | |
| end OAI | |