| import Mathlib |
| import OAI.Probability.Perceptron.Cascade.DecoratedWeightedTotal |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| namespace SphericalPerceptronFreeEnergy |
| lemma finiteCascadeLogRecursion_add_constant {X S : Type} [MeasurableSpace X] [MeasurableSpace S] |
| (ν : ProbabilityMeasure S) (step : X×S→X) (H : X→ℝ) (k : ℕ) (z : Fin k→ℝ) |
| (hz : ∀ i,0<z i) (hi : finiteCascadeFractionalIntegrable ν step H k z) (c : ℝ) (x : X) : |
| finiteCascadeLogRecursion ν step k z (fun x=>c+H x) x= |
| c+finiteCascadeLogRecursion ν step k z H x := by |
| induction k generalizing x with |
| | zero => rfl |
| | succ k ih => |
| simp only [finiteCascadeLogRecursion,ih _ (fun i=>hz i.succ) hi.2,fractionalLogMoment] |
| simp_rw [mul_add,Real.exp_add] |
| rw [integral_const_mul,Real.log_mul (Real.exp_ne_zero _) (integral_exp_pos (hi.1 x)).ne',Real.log_exp] |
| field_simp [(hz 0).ne'] |
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| lemma finiteCascadeFractionalIntegrable_add_constant {X S : Type} [MeasurableSpace X] [MeasurableSpace S] |
| (ν : ProbabilityMeasure S) (step : X×S→X) (H : X→ℝ) (k : ℕ) (z : Fin k→ℝ) |
| (hz : ∀ i,0<z i) (hi : finiteCascadeFractionalIntegrable ν step H k z) (c : ℝ) : |
| finiteCascadeFractionalIntegrable ν step (fun x=>c+H x) k z := by |
| induction k with |
| | zero => trivial |
| | succ k ih => |
| refine ⟨?_,ih _ (fun i=>hz i.succ) hi.2⟩ |
| intro x |
| simp only [finiteCascadeLogRecursion_add_constant ν step H k _ (fun i=>hz i.succ) hi.2 c,mul_add,Real.exp_add] |
| exact (hi.1 x).const_mul _ |
| end SphericalPerceptronFreeEnergy |
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| end |
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| end OAI |
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