| import Mathlib |
| import OAI.Probability.SKBarriers.Hierarchy.HierarchyBlocks |
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|
| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
| namespace SK.Analytic |
| attribute [local instance 1900] cascadeNormedGroup cascadeNormedSpace |
| attribute [local instance 2000] parameterNormedGroup parameterNormedSpace |
|
|
| theorem parameterAppend_noise_axis (a b : ℕ) (i : Fin b) : |
| parameterAppend a b (parameterNoise (E := ParameterSpace a) b (coordinateAxis b i)) = |
| coordinateAxis (a+b) ⟨a+i.val,by omega⟩ := by |
| induction b with |
| | zero => exact Fin.elim0 i |
| | succ b ih => |
| refine Fin.lastCases ?_ (fun i => ?_) i |
| · rw [coordinateAxis_last] |
| have he : (⟨a+(Fin.last b).val,by omega⟩ : Fin (a+(b+1))) = Fin.last (a+b) := by |
| apply Fin.ext; simp |
| rw [he] |
| change _ = coordinateAxis ((a+b)+1) (Fin.last (a+b)) |
| rw [coordinateAxis_last] |
| change (parameterAppend a b (parameterNoise (E := ParameterSpace a) b 0),1) = (0,1) |
| rw [map_zero,map_zero] |
| · rw [coordinateAxis_castSucc] |
| have he : (⟨a+i.castSucc.val,by omega⟩ : Fin (a+(b+1))) = (⟨a+i.val,by omega⟩ : Fin (a+b)).castSucc := by |
| rfl |
| rw [he] |
| change _ = coordinateAxis ((a+b)+1) (⟨a+i.val,by omega⟩ : Fin (a+b)).castSucc |
| rw [coordinateAxis_castSucc] |
| change (parameterAppend a b (parameterNoise (E := ParameterSpace a) b (coordinateAxis b i)),0) = _ |
| rw [ih] |
|
|
| theorem parameterBlocks_noise_axis (a b d : ℕ) (i : Fin b) : |
| parameterBlocks a b d (cascadeLift d (parameterNoise (E := ParameterSpace a) b (coordinateAxis b i))) = |
| coordinateAxis (a+(b+d)) ⟨a+i.val,by omega⟩ := by |
| induction d with |
| | zero => exact parameterAppend_noise_axis a b i |
| | succ d ih => |
| change (parameterBlocks a b d (cascadeLift d (parameterNoise (E := ParameterSpace a) b (coordinateAxis b i))),0) = _ |
| rw [ih] |
| change _ = coordinateAxis ((a+(b+d))+1) (⟨a+i.val,by omega⟩ : Fin (a+(b+d))).castSucc |
| rw [coordinateAxis_castSucc] |
| end SK.Analytic |
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| end |
| end |
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| end |
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| end OAI |
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