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a2ee19a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 | import Mathlib
import OAI.Probability.Perceptron.Cascade.DecoratedWeightedTotal
namespace OAI
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']
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
end
end OAI
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