Commit 74a0b452 authored by Dennis Müller's avatar Dennis Müller

cleaned up

parent 3c5019fd
namespace http://mathhub.info/MitM/smglom/buggy ❚
import base http://mathhub.info/MitM/Foundation ❚
import ts http://mathhub.info/MitM/smglom/typedsets ❚
import ar http://mathhub.info/MitM/smglom/arithmetics ❚
theory buggy : base:?Logic =
include ts:?AllSets ❙
include ar:?RealArithmetics ❙
possibleBug : ℝ ⟶ ℝ ⟶ ℝ ❘ = [x , y] subtraction x y ❙
possibleBug2 : ℝ ⟶ ℝ ⟶ ℝ ❘ = [x , y] x - y ❙
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namespace http://mathhub.info/MitM/smglom/DescriptionOperators ❚
import base http://mathhub.info/MitM/Foundation ❚
import sub http://gl.mathhub.info/MMT/LFX/Subtyping ❚
theory HilbertEpsilon : base:?Logic =
Eps : {A : type} (A ⟶ bool) ⟶ A ❘# ε 2❙
epsIntro : {A : type , P : A ⟶ bool} ⊦ (∃[x : A] P x) ⟶ ⊦ P (ε P) ❙
// epsIntro2 : {A : type , P : A ⟶ bool, Q : A ⟶ bool} ⊦ (∃ [x : A ] P x) ⟶ ⊦ (∀ [x : A] P x ⟶ Q x) ⟶ ε Q ❙
epsIntro3 : {A : type , P : A ⟶ bool, Q : A ⟶ bool} ⊦ (∃[x : A] P x) ⟶ ({x : A} ⊦ P x ⟶ ⊦ Q x) ⟶ ⊦ Q (ε P) ❙
theory PeonoIota : base:?Logic =
include sub:?LFSubtyped❙
include base:?Math ❙
bla : type ❘= ⟨ ℕ | [x] ⊦ x ≐ 0⟩ ❙
blu = PredOf bla ❙
namespace http://mathhub.info/MitM/smglom/Functions ❚
import ts http://mathhub.info/MitM/smglom/typedsets ❚
import base http://mathhub.info/MitM/Foundation ❚
theory Functions : base:?Logic =
structure FunctionRelations ts:?Relations ❙
\ No newline at end of file
namespace http://mathhub.info/MitM/smglom/HilbertChoice ❚
import base http://mathhub.info/MitM/Foundation ❚
theory HilbertChoice : base:?Logic =
Eps : {A : type} (bool ⟶ A) ⟶ A ❘# ε 2❙
\ No newline at end of file
......@@ -86,8 +86,9 @@ theory RealArithmetics : base:?Logic =
theory ExtendedReals : base:?Logic =
include ?RealArithmetics ❙
extended_reals : type ❘ # ℝ∞ ❙
reals_are_extended : ℝ <* ℝ∞ ❙
infty : ℝ∞ ❘ # ∞ prec 1 ❙
// extended_reals : type ❘ # ℝ∞ ❙
// reals_are_extended : ℝ <* ℝ∞ ❙
infty : ℝ ❘ # ∞ prec 1 ❙
// leq : ℝ∞ ⟶ ℝ∞ ⟶ bool ❘ # 1 ≤ 2 prec 101 ❙
// TODO neginfty : ℝ∞ ❘ = - ∞ ❘ # -∞ ❙
namespace http://mathhub.info/MitM/smglom/FinSequences
namespace http://mathhub.info/MitM/smglom ❚
import base http://mathhub.info/MitM/Foundation ❚
import arith http://mathhub.info/MitM/smglom/arithmetics ❚
import seq http://cds.omdoc.org/urtheories❚
import sig http://gl.mathhub.info/MMT/LFX/Sigma ❚
theory FinSequences : base:?Logic =
include arith:?NaturalArithmetics ❙
below : ℕ ⟶ type ❙
toBelow : {n : ℕ} below (succ n) ❘# toB 1❙
succBelow : {n : ℕ} below n ⟶ below (succ n) ❘# succB 2❙
toBelow : {n : ℕ} below (Succ n) ❘# toB 1❙
succBelow : {n : ℕ} below n ⟶ below (Succ n) ❘# succB 2❙
belowIsSubtypeOfNat : {n : ℕ} (below n) <* ℕ ❙
belowToNat : {n : ℕ} below (succ n) ⟶ ℕ ❘# btn 1 2❙
belowToNat : {n : ℕ} below (Succ n) ⟶ ℕ ❘# btn 1 2❙
belowToNatBaseCase : {m : ℕ} ⊦ belowToNat m (toBelow m) ≐ m❙
belowToNatRecCase : {m : ℕ, b : below (succ m)} ⊦ belowToNat (succ m) (succBelow (succ m) b) ≐ belowToNat m b ❙
belowToNatRecCase : {m : ℕ, b : below (Succ m)} ⊦ belowToNat (Succ m) (succBelow (Succ m) b) ≐ belowToNat m b ❙
belowToNatExample : ⊦ btn 3 (toB 3) ≐ 3❙
belowToNatExample2 : ⊦ btn 4 (succB (toB 3)) ≐ 3 ❙
belwoToNatExample2_2 : ⊦ belowToNat (succ 3) (succBelow (succ 3) (toBelow 3)) ≐ 3❙
belowToNatExample : ⊦ btn 3 (toB 3) ≐ 3❙
belowToNatExample2 : ⊦ btn 4 (succB (toB 3)) ≐ 3 ❙
belwoToNatExample2_2 : ⊦ belowToNat (Succ 3) (succBelow (Succ 3) (toBelow 3)) ≐ 3❙
// belowToNat2 : {n : ℕ} below (Succ n) ⟶ ℕ ❘= [n, b] n ❘# btn2 %I1 ❙
// belowToNat3 : {n : ℕ, m : ℕ} below n ⟶ ⊦ m < n ⟶ ℕ ❘= [n , m , b, p ] m ❘# btn3 %I1 ❙
......
namespace http://mathhub.info/MitM/smglom/measures ❚
namespace http://mathhub.info/MitM/smglom/calculus ❚
import base http://mathhub.info/MitM/Foundation ❚
import ts http://mathhub.info/MitM/smglom/typedsets ❚
import seq http://cds.omdoc.org/urtheories ❚
import supr http://mathhub.info/MitM/smglom/supremum ❚
import arith http://mathhub.info/MitM/smglom/arithmetics ❚
import ms http://mathhub.info/MitM/smglom/measures ❚
theory LebesgueIntegral : base:?Logic =
include ?Measure ❙
include ms:?Measure ❙
include ts:?Functions ❙
include ts:?SetCollection ❙
include ?SimpleFunctions ❙
include ts:?SetSum ❙
include supr:?SupremumReal ❙
include ?LebesgueMeasurable ❙
include arith:?RealArithmetics ❙
include ts:?SupremumReal ❙
include ms:?LebesgueMeasurable ❙
// include arith:?RealArithmetics ❙
simpleLebesgueIntegral : {m : measureSpace , s : nonnegativeSimpleFunction m } ℝ
❘ = [m , s] setSum ((im (πl s) (fullset (m.universe))) : set ℝ)
(theorem_nonnegativeSimpleFunctionHasFiniteValues m s) (([x] x ⋅ (m.measure.measure) (elem (m.sigma_algebra.coll) (preim (πl s) (single x)))) : ℝ ⟶ ℝ) ❙
(theorem_nonnegativeSimpleFunctionHasFiniteValues m s) (([x] x ⋅ (m.rmeasure.measure) (elem (m.sigma_algebra.coll) (preim (πl s) (single x)))) : ℝ ⟶ ℝ) ❙
upperFunction : {a : type} (a ⟶ ℝ) ⟶ a ⟶ ℝ ❘ = [a , f , x ] max (f x) 0❙
......@@ -76,13 +74,4 @@ theory LebesgueIntegral : base:?Logic =
theory lebesgueIntegralAlt : base:?Logic =
include ?LebesgueIntegral ❙
computationalLebesgueIntegral ❙
theory LebesgueIntervalIntegral : base:?Logic =
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namespace http://mathhub.info/MitM/smglom/measures ❚
namespace http://mathhub.info/MitM/smglom/calculus ❚
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❚
import top http://mathhub.info/MitM/smglom/topology ❚
import ms http://mathhub.info/MitM/smglom/measures ❚
theory SimpleFunctions : base:?Logic =
include ?Measure ❙
include ms:?Measure ❙
include ts:?Functions ❙
include ?PowerSigmaAlgebra ❙
include ?BorelMeasurable ❙
include ms:?PowerSigmaAlgebra ❙
include ms:?BorelMeasurable ❙
// include top:?RealTopology ❙
......
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 seq http://cds.omdoc.org/urtheories ❚
import supr http://mathhub.info/MitM/smglom/supremum ❚
theory BochnerIntegral : base:?Logic =
bochnerIntegrable ❙
\ No newline at end of file
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❚
import calc http://mathhub.info/MitM/smglom/calculus ❚
theory RiemannIntegrable : base:?Logic =
\ No newline at end of file
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 ❙
\ No newline at end of file
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❚
import calc http://mathhub.info/MitM/smglom/calculus ❚
theory OuterMeasure : base:?Logic =
include ts:?SetCollection ❙
theory outerMeasure_theory : base:?Logic =
measure ❙
emptyIsZero
\ No newline at end of file
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❚
import calc http://mathhub.info/MitM/smglom/calculus ❚
theory RiemannIntegrable : base:?Logic =
include calc:?Intervals ❙
include ?AlmostEverywhere ❙
prop_lebesgueCriterion ❙
\ No newline at end of file
......@@ -95,3 +95,17 @@ theory CountingMeasure : base:?Logic =
countingMeasureSpace : type ⟶ measureSpace ❘ // slow! = [A] ['universe := A , sigma_algebra := powerSigmaAlgebra A , measure := countingMeasure A (powerSigmaAlgebra A) '] ❙
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❙
namespace http://mathhub.info/MitM/smglom/probability ❚
import base http://mathhub.info/MitM/Foundation ❚
theory AI1 : base:?Logic =
......@@ -22,13 +22,12 @@ theory ProbabilityMeasure : base:?Logic =
theory ProbabilitySpaceTheorems : base:?Logic =
include ?ProbabilityMeasure ❙
include measures?LebesgueIntegral ❙
include calculus?LebesgueIntegral ❙
include measures?Measurable ❙
include measures?LebesgueMeasurable ❙
theory probabilitySpaceTheorems_theory : base:?Logic > PS : probabilitySpace ❘ =
include measures?LebesgueIntegral ❙
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) ❙
......@@ -39,17 +38,17 @@ theory ProbabilitySpaceTheorems : base:?Logic =
// variance : Ω ⟶ ℝ ❙
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)
randomVariable : {A : type} ⟨ t : (((PS.universe) ⟶ A) × sigmaAlgebra A) | ⊦ isRandomVariable A (πl t) (πr t) ⟩
isRealValuedRandomVariable : ((PS.universe) ⟶ ℝ) ⟶ bool ❘= [f] ❙
realValuedRandomVariable = Σ f: ((PS.universe) ⟶ ℝ) . ⊦ isRealValuedRandomVariable f ❘# ℝRv ❙
isRealValuedRandomVariable : ((PS.universe) ⟶ ℝ) ⟶ bool ❘// = [f] ❙
realValuedRandomVariable = ⟨ f: ((PS.universe) ⟶ ℝ) | ⊦ isRealValuedRandomVariable f ⟩ ❘# ℝRv ❙
// theorems ❙
lemma_probabilityLe1 : ⊦ ∀[x : set Ω] probability x ≤ 1 ❙
// lemma_probabilityLe1 : ⊦ ∀[x : set Ω] probability x ≤ 1 ❙
lemma_omegaNotEmpty : ⊦ fullset Ω ≠ ∅ ❙
lemma_probabilityGe1Eq1 : {x : set Ω} ⊦ 1 ≤ probability x ⟶ ⊦ probability x ≐ 1 ❙
// lemma_probabilityGe1Eq1 : {x : set Ω} ⊦ 1 ≤ probability x ⟶ ⊦ probability x ≐ 1 ❙
......@@ -57,9 +56,5 @@ theory ProbabilitySpaceTheorems : base:?Logic =
theory bla : base:?Logic =
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 ❘ ❙
\ No newline at end of file
......@@ -4,7 +4,8 @@ import base http://mathhub.info/MitM/Foundation ❚
import sig http://gl.mathhub.info/MMT/LFX/Sigma ❚
theory FiniteSets : sig:?LFSigma =
theory FiniteSets : base:?Logic =
include base:?ProductType ❙
include ?AllSets ❙
......
namespace http://mathhub.info/MitM/smglom/typedsets ❚
import base http://mathhub.info/MitM/Foundation ❚
import ar http://mathhub.info/MitM/smglom/arithmetics ❚
theory PrOSet : base:?Logic =
include ?Relations ❙
......@@ -81,4 +82,10 @@ theory POSet : base:?Logic =
poset = Mod poset_theory ❘ role abbreviation ❙
theory SupremumReal : base:?Logic =
include ?AllSets ❙
include ?LeastMost ❙
include ar:?ExtendedReals ❙
supremumReal : set ℝ ⟶ ℝ❘= [s] least ([a] ∀[x] x ∈ s ⇒ x ≤ a) ([a,b] a ≤ b) ❘# supR 1 ❙
......@@ -6,8 +6,7 @@ import sig http://gl.mathhub.info/MMT/LFX/Sigma ❚
import ar http://mathhub.info/MitM/smglom/arithmetics ❚
theory SetSum : base:?Logic =
include ar:?RealArithmetics ❙
include sig:?LFSigma ❙
include ar:?RealArithmetics ❙
include ts:?FiniteSets ❙
......
namespace http://mathhub.info/MitM/smglom/typedsets ❚
import base http://mathhub.info/MitM/Foundation ❚
import sig http://gl.mathhub.info/MMT/LFX/Sigma
import arith http://mathhub.info/MitM/smglom/arithmetics
theory TypedSets : base:?Logic =
include base:?Lists ❙
......@@ -129,7 +129,14 @@ theory SetStructures : base:?Logic =
dom : setstruct ⟶ type ❘ = [S] (S.universe) ❘ role abbreviation❙
doms : {S : setstruct}set (dom S) ❘ = [S] fullset (dom S) ❘ role abbreviation ❙
theory IndicatorFunction : base:?Logic =
include ?TypedSets ❙
include arith:?NaturalNumbers ❙
include base:?DescriptionOperator ❙
indicatorFunction : {A : type} set A ⟶ A ⟶ ℕ ❘= [A,s,a] if (a ∈ s) then (1 : ℕ) else 0 ❘ ❙
theory AllSets : base:?Logic =
include ?FiniteCardinality ❙
......@@ -160,7 +167,7 @@ theory SetCollection : base:?Logic =
theory TaggedSetCollection : base:?Logic =
include ?SetCollection ❙
include sig:?LFSigma
include base:?ProductType
taggedCollection : type ⟶ type ❘= [G] set (G × set G) ❙
......
namespace http://mathhub.info/MitM/smglom/supremum❚
import base http://mathhub.info/MitM/Foundation ❚
import ts http://mathhub.info/MitM/smglom/typedsets ❚
import ar http://mathhub.info/MitM/smglom/arithmetics ❚
theory SupremumReal : base:?Logic =
include ts:?AllSets ❙
include ts:?LeastMost ❙
include ar:?ExtendedReals ❙
supremumReal : set ℝ ⟶ ℝ ❘= [s] least ([a] ∀[x] x ∈ s ⇒ x ≤ a) ([a,b] a ≤ b) ❘# supR 1 ❙
\ No newline at end of file
namespace http://mathhub.info/MitM/smglom/topology ❚
import base http://mathhub.info/MitM/Foundation ❚
import calc http://mathhub.info/MitM/smglom/calculus ❚
theory RealTopology : base:?Logic =
include calc:?RealMetricSpace❙
include calc:?MetricSpaceAsTopology ❙
include ?OpenSetTopology ❙
realTopology = metricAsTopology realmetricspace ❘ : topology ℝ❙
realTopologicalSpace = [' universe := ℝ , topology := realTopology '] ❘: topologicalSpace ❙
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