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c4cdbbd | 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 | import OAI.Probability.IsingPerceptron.NoiseTreeKeep
namespace OAI
/-! A finite tree uses only the updates before its depth. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
namespace InvariantIsing
lemma noiseTreeKeep_update_congr {A S : Type} [MeasurableSpace A] [MeasurableSpace S] [Nonempty A]
(n : ℕ) (b : ℕ → ℝ) (μ : ℕ → ProbabilityMeasure A)
(c : ℕ → S × A → ℝ) (u u' : ℕ → S × A → S)
(hu : ∀ i < n, u i = u' i) (s : S) (T : NoiseTree A n) :
noiseTreeKeep n b μ c u s T = noiseTreeKeep n b μ c u' s T := by
induction n generalizing b μ c u u' s with
| zero => rfl
| succ n ih =>
unfold noiseTreeKeep
congr 1
funext p
apply congrArg (fun V => (c 0 (s,p.2.1)*p.1,(p.2.1,V)))
rw [hu 0 (by omega)]
exact ih _ _ _ _ _ (fun i hi => hu (i+1) (by omega)) _ _
end InvariantIsing
end
end OAI
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