| import Mathlib |
| import OAI.Probability.SKBarriers.Locking.NarrowRetainedFactor |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Set |
| namespace SK.Analytic |
| section Scale |
| variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] |
|
|
| theorem vectorIncrementChain_div_scale (l : List (ℝ × E)) (c : ℝ) (hc : c≠0) (f : E → ℝ) : |
| vectorIncrementChain (l.map (fun p => (p.1/c,p.2))) (fun x => c*f x)= |
| fun x => c*vectorIncrementChain l f x := by |
| induction l with |
| | nil => rfl |
| | cons p l ih => |
| simp only [List.map_cons,vectorIncrementChain,ih,vectorStep_div_scale p.1 c hc] |
|
|
| theorem vectorIncrementAverage_div_scale (l : List (ℝ × E)) (c : ℝ) (hc : c≠0) (f g : E → ℝ) : |
| vectorIncrementAverage (l.map (fun p => (p.1/c,p.2))) (fun x => c*f x) g= |
| vectorIncrementAverage l f g := by |
| induction l with |
| | nil => rfl |
| | cons p l ih => |
| simp only [List.map_cons,vectorIncrementAverage,ih,vectorIncrementChain_div_scale l c hc, |
| vectorStepAverage_div_scale p.1 c hc] |
| end Scale |
|
|
| def narrowCommonEmbedding : (ℝ × ℝ) →L[ℝ] NarrowRetainedState := |
| ((ContinuousLinearMap.fst ℝ ℝ ℝ).prod 0).prod (ContinuousLinearMap.id ℝ (ℝ × ℝ)) |
|
|
| @[simp] theorem narrowCommonEmbedding_apply (p : ℝ × ℝ) : narrowCommonEmbedding p=((p.1,0),p) := rfl |
|
|
| theorem narrowCommonBase_comp (f : ℝ → ℝ) : |
| (fun p : NarrowRetainedState => f p.1.1+2*f p.2.1) ∘ narrowCommonEmbedding= |
| fun p => 3*f p.1 := by |
| funext p |
| simp only [Function.comp_def,narrowCommonEmbedding_apply] |
| ring |
|
|
| theorem narrowCommon_image (c : List (ℝ × (ℝ × ℝ))) : |
| (c.map (fun p => (p.1/3,p.2))).map (fun p => (p.1,narrowCommonEmbedding p.2))=narrowRetainedCommon c := by |
| simp only [List.map_map,narrowRetainedCommon,Function.comp_def,narrowCommonEmbedding_apply,Prod.mk.eta] |
|
|
| theorem narrowRetainedCommon_value (c : List (ℝ × (ℝ × ℝ))) (f : ℝ → ℝ) (p : ℝ × ℝ) : |
| vectorIncrementChain (narrowRetainedCommon c) (fun p : NarrowRetainedState => f p.1.1+2*f p.2.1) |
| (narrowCommonEmbedding p)=3*scalarIncrementChain (weightedUnderlying c) f p.1 := by |
| have H := congrFun (vectorIncrementChain_pullback narrowCommonEmbedding |
| (c.map (fun p => (p.1/3,p.2))) (fun p : NarrowRetainedState => f p.1.1+2*f p.2.1)) p |
| rw [narrowCommonBase_comp,narrowCommon_image,vectorIncrementChain_div_scale c 3 (by norm_num),weightedBranch_value] at H |
| exact H.symm |
|
|
| theorem narrowRetainedCommon_average (c : List (ℝ × (ℝ × ℝ))) (f : ℝ → ℝ) |
| (g : NarrowRetainedState → ℝ) (p : ℝ × ℝ) : |
| vectorIncrementAverage (narrowRetainedCommon c) (fun p : NarrowRetainedState => f p.1.1+2*f p.2.1) g |
| (narrowCommonEmbedding p)=vectorIncrementAverage c (fun p : ℝ × ℝ => f p.1) (g ∘ narrowCommonEmbedding) p := by |
| have H := congrFun (vectorIncrementAverage_pullback narrowCommonEmbedding |
| (c.map (fun p => (p.1/3,p.2))) (fun p : NarrowRetainedState => f p.1.1+2*f p.2.1) g) p |
| rw [narrowCommonBase_comp,narrowCommon_image,vectorIncrementAverage_div_scale c 3 (by norm_num)] at H |
| exact H.symm |
|
|
| end SK.Analytic |
|
|
| end |
| end |
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| end OAI |
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|
