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135052a | 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 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 | 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
end OAI
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