Commit fcc516d1 authored by SuperDuckDuck's avatar SuperDuckDuck

added stuff

parent 7a49c6b8
namespace http://mathhub.info/MitM/smglom/arithmetics ❚
import base http://mathhub.info/MitM/Foundation ❚
// used as a return type in for the indicator function in indicatorFunction.mmt❚
theory ZeroOne : base:?Logic =
include ?NaturalNumbers ❙
zeroOne : type ❘= ⟨ ℕ | [x] ⊦ x ≐ 1 ∨ x ≐ 0⟩ ❙
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namespace http://mathhub.info/MitM/smglom/measures ❚
import base http://mathhub.info/MitM/Foundation ❚
import ts http://mathhub.info/MitM/smglom/typedsets ❚
import ar http://mathhub.info/MitM/smglom/arithmetics ❚
theory NullSets : base:?Logic =
include ?Measure ❙
isNullSet : {m : measureSpace} set (m.universe) ⟶ bool ❘// slow! = [m , s] (s ∈ m.sigma_algebra.coll) ∧ (((m.measure.measure)
(elem (m.sigma_algebra.coll) s)) ≐ 0)❙
allNullSets : {m : measureSpace} collection (m.universe) ❘// slow!= [m] ⟪ (m.sigma_algebra.coll) | ([s] (((m.measure.measure)
(elem (m.sigma_algebra.coll) s)) ≐ 0))⟫ ❙
theory AlmostEverywhere : base:?Logic =
include ?NullSets ❙
almostEverywhere : {m : measureSpace} ((m.universe) ⟶ bool) ⟶ bool ❘
= [m,P] ∃ [n] n ∈ (allNullSets m) ∧ (∀[x] x ∈ (fullset (m.universe) \ n) ⇒ P x)❘# AE 1 2❙
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namespace http://mathhub.info/MitM/smglom/measures ❚
import base http://mathhub.info/MitM/Foundation ❚
import ts http://mathhub.info/MitM/smglom/typedsets ❚
import lfx http://gl.mathhub.info/MMT/LFX/Sigma ❚
import exr http://mathhub.info/MitM/smglom/arithmetics ❚
import seq http://cds.omdoc.org/urtheories ❚
import desc http://mathhub.info/MitM/smglom/DescriptionOperators❚
theory LebesgueMeasure : base:?Logic =
include ?Measure ❙
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......@@ -47,9 +47,10 @@ theory Measure : base:?Logic =
measurableSpace = {'universe : type , sigma_algebra : sigmaAlgebra universe '} ❙
// toMeasurableSpace : measureSpace ⟶ measurableSpace ❘ = [ms] ['universe := ms.universe , sigma_algebra := ms.sigma_algebra'] ❙
// toMeasureSpace : {ms: measurableSpace} measure (ms.universe) (ms.sigma_algebra) ⟶ measureSpace
// very slow! toMeasureSpace : {ms: measurableSpace} measure (ms.universe) (ms.sigma_algebra) ⟶ measureSpace
❘// = [ms, m] ['universe := ms.universe , sigma_algebra := ms.sigma_algebra , measure := m'] ❙
isFiniteMeasure : measureSpace ⟶ bool ❘// slow! = [ms] (ms.measure.measure) (elem (ms.sigma_algebra.coll) (fullset (ms.universe))) < ∞❙
......
namespace http://mathhub.info/MitM/smglom/probability ❚
import base http://mathhub.info/MitM/Foundation ❚
import sub http://gl.mathhub.info/MMT/LFX/Subtyping ❚
// import sub http://gl.mathhub.info/MMT/LFX/Subtyping ❚
theory ProbabilityMeasure : base:?Logic =
include measures?Measure ❙
......@@ -22,6 +22,9 @@ theory ProbabilityMeasure : base:?Logic =
theory ProbabilitySpaceTheorems : base:?Logic =
include ?ProbabilityMeasure ❙
include measures?LebesgueIntegral ❙
include measures?Measurable ❙
include measures?LebesgueMeasurable ❙
theory probabilitySpaceTheorems_theory : base:?Logic > PS : probabilitySpace ❘ =
......@@ -29,17 +32,18 @@ theory ProbabilitySpaceTheorems : base:?Logic =
omega = (PS.universe) ❘ # Ω ❙
probability : (typeof (set (PS.universe)) (PS.sigma_algebra.coll)) ⟶ ℝ ❘ // slow! = (PS.measure.measure) ❙
events : collection (PS.universe) ❘ // sometimes slow! = (PS.sigma_algebra.coll) ❙
currentMeasurespace : measureSpace ❘ # mspace ❙
// eventsSigmaAlgebra = (PS.sigma_algebra) ❙
// probabilityMeasure : measure Ω (eventsSigmaAlgebra) ❙
expectation : quasiIntegrableFunction mspace ⟶ ℝ ❙
variance : lebesgueIntegrableFunction mspace ⟶ ℝ ❙
// expectation : Ω ⟶ ⊦ quasiIntegrable ⟶ ℝ ❙
// variance : Ω ⟶ ℝ ❙
isRandomVariable : {A : type} ((PS.universe) ⟶ A) ⟶ sigmaAlgebra A ⟶ bool ❙
// slow ! randomVariable : {A : type} Σ t : (((PS.universe) ⟶ A) × sigmaAlgebra A) . ⊦ isRandomVariable A (πl t) (πr t) ❙
isRandomVariable : {A : type} ((PS.universe) ⟶ A) ⟶ sigmaAlgebra A ⟶ bool
randomVariable : {A : type} Σ t : (((PS.universe) ⟶ A) × sigmaAlgebra A) . ⊦ isRandomVariable A (πl t) (πr t) ❙
isRealValuedRandomVariable : ((PS.universe) ⟶ ℝ) ⟶ bool ❙
realValuedRandomVariable = Σ f: ((PS.universe) ⟶ ℝ) . ⊦ isRealValuedRandomVariable f ❙
isRealValuedRandomVariable : ((PS.universe) ⟶ ℝ) ⟶ bool ❘= [f]
realValuedRandomVariable = Σ f: ((PS.universe) ⟶ ℝ) . ⊦ isRealValuedRandomVariable f ❘# ℝRv
// theorems ❙
......@@ -48,5 +52,14 @@ theory ProbabilitySpaceTheorems : base:?Logic =
lemma_probabilityGe1Eq1 : {x : set Ω} ⊦ 1 ≤ probability x ⟶ ⊦ probability x ≐ 1 ❙
theory bla : base:?Logic =
namespace http://mathhub.info/MitM/smglom/probability ❚
import base http://mathhub.info/MitM/Foundation ❚
import sets http://mathhub.info/MitM/smglom/typedsets ❚
import seqs http://mathhub.info/MitM/smglom/calculus ❚
// theory Vitali : base:?Logic =
include sets:?AllSets ❙
include seqs:?Sequences ❙
vitaliSet : set ℕ❙
theory VitaliCoinToss : base:?Logic =
include sets:?TypedSets ❙
include seqs:?Sequences ❙
include base:?DescriptionOperator ❙
vitaliSet : set (sequence ℕ) ❘= ⟪ (fullset (ℕ ⟶ ℕ)) | ([f] ∀[i] f i ≐ 0 ∨ f i ≐ 1) ⟫ ❙
flipOneZero : ℕ ⟶ ℕ ❘= [n] (if (n ≐ 1) then 0 else (if (n ≐ 0) then (1 : ℕ) else n))❙
flipAt : (sequence ℕ) ⟶ ℕ ⟶ (sequence ℕ) ❘= [oldSeq, n] [i : ℕ] (if (i ≐ n) then (flipOneZero (oldSeq i)) else (oldSeq i)) ❙
// vitaliTheorem : ⊦ (¬ (∃[P] )) ❙
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namespace http://mathhub.info/MitM/smglom/typedsets ❚
import base http://mathhub.info/MitM/Foundation ❚
import arith http://mathhub.info/MitM/smglom/arithmetics ❚
theory IndicatorFunction : base:?Logic =
include ?AllSets ❙
include arith:?NaturalNumbers ❙
indicatorFunction : {A : type} set A ⟶ A ⟶ ℕ ❘= [A,s,a] if (a ∈ s) then (1 : ℕ) else 0 ❘ ❙
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