| 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 | |