Constants: mtSet

denition mtSet::(⁰A,⁰α::null)Set (Set{})

where Set{} ≡(λ τ . Abs-Setbasexx{}::⁰αsetyy)

lemma mtSet-dened[simp,code-unfold]:δ(Set{}) =true hproofi

lemma mtSet-valid[simp,code-unfold]:υ(Set{}) =true hproofi

lemma mtSet-rep-set:ppRep-Setbase (Set{}τ)qq={}

hproofi

lemma[simp,code-unfold]:const Set{}

hproofi

Note that the collection types in OCL allow for null to be included; however, there is the null-collection into which inclusion yields invalid.

1.17.5. Denition: Including

denition OclIncluding :: [(⁰A,⁰α::null) Set,(⁰A,⁰α)val]⇒(⁰A,⁰α)Set where OclIncluding x y = (λ τ .if (δ x)τ =true τ ∧(υ y)τ =true τ

then Abs-Setbasexx ppRep-Setbase (x τ)qq ∪ {y τ}yy else invalid τ )

notation OclIncluding (-−>includingS et0(-⁰))

interpretation OclIncluding :prole-bind-v OclIncluding λx y.Abs-SetbasexxppRep-Setbase xqq∪ {y}yy hproofi

syntax

-OclFinset ::args =>(⁰A,⁰a::null)Set (Set{(-)}) translations

Set{x,xs}==CONST OclIncluding (Set{xs}) x Set{x} ==CONST OclIncluding (Set{})x

1.17.6. Denition: Excluding

denition OclExcluding :: [(⁰A,⁰α::null)Set,(⁰A,⁰α)val]⇒(⁰A,⁰α)Set where OclExcluding x y = (λ τ . if (δx) τ =true τ ∧ (υ y)τ =true τ

then Abs-Setbasexx ppRep-Setbase(x τ)qq− {y τ}yy else ⊥)

notation OclExcluding (-−>excludingS et0

(-⁰))

lemma OclExcluding-inv: (x::Set(⁰b::{null}))6=⊥=⇒x 6=null =⇒ y 6=⊥ =⇒ xxppRep-Setbasexqq− {y}yy∈ {X.X =bot ∨X =null ∨(∀x∈ppXqq.x 6=bot)}

hproofi

interpretation OclExcluding :prole-bind-vOclExcluding λx y.Abs-SetbasexxppRep-Setbasexqq− {y}yy hproofi

1.17.7. Denition: Includes

denition OclIncludes :: [(⁰A,⁰α::null)Set,(⁰A,⁰α)val]⇒ ⁰ABoolean where OclIncludes x y = (λ τ . if (δx)τ =true τ ∧ (υ y)τ =true τ

then xx(y τ)∈ppRep-Setbase (x τ)qq yy else ⊥ )

notation OclIncludes (-−>includesS et0

(-⁰) )

interpretation OclIncludes :prole-bind-vOclIncludes λx y.xxy ∈ppRep-Setbase xqqyy hproofi

1.17.8. Denition: Excludes

denition OclExcludes :: [(⁰A,⁰α::null)Set,(⁰A,⁰α) val]⇒ ⁰ABoolean where OclExcludes x y = (not(OclIncludes x y))

notation OclExcludes (-−>excludesS et0

(-⁰) )

The case of the size denition is somewhat special, we admit explicitly in Featherweight OCL the possibility of innite sets. For the size denition, this requires an extra condition that assures that the cardinality of the set is actually a dened integer.

interpretation OclExcludes :prole-bind-v OclExcludesλx y.xxy ∈/ppRep-Setbasexqqyy hproofi

1.17.9. Denition: Size

denition OclSize :: (⁰A,⁰α::null)Set ⇒ ⁰AInteger

where OclSize x = (λ τ . if (δ x)τ =true τ ∧nite(ppRep-Setbase(x τ)qq) then xxint(card ppRep-Setbase (x τ)qq)yy

else ⊥) notation

OclSize (-−>sizeS et0

(⁰) )

The following denition follows the requirement of the standard to treat null as neutral element of sets.

It is a well-documented exception from the general strictness rule and the rule that the distinguished argument self should be non-null.

1.17.10. Denition: IsEmpty

denition OclIsEmpty :: (⁰A,⁰α::null)Set ⇒ ⁰ABoolean

where OclIsEmpty x = ((υ x and not(δx))or ((OclSize x) .

=0)) notation OclIsEmpty (-−>isEmptyS et0

(⁰) )

1.17.11. Denition: NotEmpty

denition OclNotEmpty :: (⁰A,⁰α::null)Set ⇒ ⁰ABoolean where OclNotEmpty x = not(OclIsEmpty x)

notation OclNotEmpty (-−>notEmptyS et0

(⁰) )

1.17.12. Denition: Any

denition OclANY :: [(⁰A,⁰α::null)Set]⇒(⁰A,⁰α) val where OclANY x = (λ τ . if (υ x)τ =true τ

then if (δ x and OclNotEmpty x) τ =true τ then SOME y.y ∈ppRep-Setbase (x τ)qq else null τ

else ⊥) notation OclANY (-−>anyS et0

(⁰))

1.17.13. Denition: Forall

The denition of OclForall mimics the one of op and: OclForall is not a strict operation.

denition OclForall :: [(⁰A,⁰α::null)Set,(⁰A,⁰α)val⇒(⁰A)Boolean]⇒ ⁰ABoolean where OclForall S P = (λ τ .if (δS)τ =true τ

then if (∃x∈ppRep-Setbase (S τ)qq.P(λ-.x)τ =false τ) then false τ

else if (∃x∈ppRep-Setbase (S τ)qq.P(λ-.x) τ =invalid τ) then invalid τ

else if (∃x∈ppRep-Setbase(S τ)qq.P(λ-.x)τ =null τ) then null τ

else true τ else ⊥)

syntax

-OclForallSet :: [(⁰A,⁰α::null)Set,id,(⁰A)Boolean]⇒ ⁰ABoolean ((-)−>forAllS et0

(-|-⁰)) translations

X−>forAllS et(x |P) ==CONST UML-Set.OclForall X (%x.P)

1.17.14. Denition: Exists

Like OclForall, OclExists is also not strict.

denition OclExists :: [(⁰A,⁰α::null)Set,(⁰A,⁰α)val⇒(⁰A)Boolean]⇒ ⁰ABoolean

where OclExists S P =not(UML-Set.OclForall S (λX.not (P X))) syntax

-OclExistSet :: [(⁰A,⁰α::null)Set,id,(⁰A)Boolean]⇒ ⁰ABoolean ((-)−>existsS et0

(-|-⁰)) translations

X−>existsS et(x |P) ==CONST UML-Set.OclExists X (%x.P)

1.17.15. Denition: Iterate

denition OclIterate :: [(⁰A,⁰α::null)Set,(⁰A,⁰β::null)val,

(⁰A,⁰α)val⇒(⁰A,⁰β)val⇒(⁰A,⁰β)val]⇒(⁰A,⁰β)val

where OclIterate S A F = (λ τ . if (δ S) τ =true τ ∧(υ A) τ =true τ ∧ niteppRep-Setbase (S τ)qq then (Finite-Set.fold (F) (A) ((λa τ .a) ` ppRep-Setbase(S τ)qq))τ

else ⊥) syntax

-OclIterateSet :: [(⁰A,⁰α::null)Set,idt,idt, ⁰α, ⁰β] =>(⁰A,⁰γ)val (-−>iterateS et0

(-;-=-|-⁰) ) translations

X−>iterateS et(a;x =A |P) ==CONST OclIterate X A(%a.(%x.P))

1.17.16. Denition: Select

denition OclSelect :: [(⁰A,⁰α::null)Set,(⁰A,⁰α)val⇒(⁰A)Boolean]⇒(⁰A,⁰α)Set where OclSelect S P = (λτ .if (δ S)τ =true τ

then if (∃x∈ppRep-Setbase (S τ)qq.P(λ-.x) τ =invalid τ) then invalid τ

else Abs-Setbasexx{x∈ppRep-Setbase (S τ)qq.P (λ-.x)τ 6=false τ}yy else invalid τ)

syntax

-OclSelectSet :: [(⁰A,⁰α::null)Set,id,(⁰A)Boolean]⇒ ⁰ABoolean ((-)−>selectS et0(-|-⁰)) translations

X−>selectS et(x |P) ==CONST OclSelect X (%x.P)

1.17.17. Denition: Reject

denition OclReject :: [(⁰A,⁰α::null)Set,(⁰A,⁰α)val⇒(⁰A)Boolean]⇒(⁰A,⁰α::null)Set where OclReject S P =OclSelect S (not o P)

syntax

-OclRejectSet :: [(⁰A,⁰α::null) Set,id,(⁰A)Boolean]⇒ ⁰ABoolean ((-)−>rejectS et0(-|-⁰)) translations

X−>rejectS et(x |P) ==CONST OclReject X (%x.P)

1.17.18. Denition: IncludesAll

denition OclIncludesAll :: [(⁰A,⁰α::null)Set,(⁰A,⁰α) Set]⇒ ⁰ABoolean where OclIncludesAll x y = (λ τ . if (δ x)τ =true τ ∧(δy)τ =true τ

then xxppRep-Setbase(yτ)qq⊆ppRep-Setbase(x τ)qq yy else ⊥ )

notation OclIncludesAll (-−>includesAllS et0

(-⁰) )

interpretation OclIncludesAll :prole-bind-dOclIncludesAll λx y.xxppRep-Setbaseyqq⊆ppRep-Setbasexqqyy hproofi

1.17.19. Denition: ExcludesAll

denition OclExcludesAll :: [(⁰A,⁰α::null) Set,(⁰A,⁰α)Set]⇒ ⁰ABoolean where OclExcludesAll x y = (λ τ . if (δx) τ =true τ ∧(δ y)τ =true τ

then xxppRep-Setbase(yτ)qq∩ppRep-Setbase (x τ)qq={}yy

else ⊥ ) notation OclExcludesAll (-−>excludesAllS et0

(-⁰) )

interpretation OclExcludesAll :prole-bind-d OclExcludesAll λx y.xxppRep-Setbase yqq∩ppRep-Setbasexqq= {}yy

hproofi

1.17.20. Denition: Union

denition OclUnion :: [(⁰A,⁰α::null)Set,(⁰A,⁰α)Set]⇒(⁰A,⁰α)Set where OclUnion x y = (λ τ . if (δx) τ =true τ ∧ (δ y) τ =true τ

then Abs-SetbasexxppRep-Setbase (y τ)qq∪ ppRep-Setbase(x τ)qq yy else ⊥ )

notation OclUnion (-−>unionS et0(-⁰) )

lemma OclUnion-inv: (x::Set(⁰b::{null}))6=⊥=⇒x 6=null =⇒ y 6=⊥ =⇒y 6=null =⇒ xxppRep-Setbase yqq∪ppRep-Setbase xqqyy∈ {X.X =bot ∨X =null ∨(∀x∈ppXqq.x 6=bot)}

hproofi

interpretation OclUnion :prole-bind-dOclUnion λx y.Abs-SetbasexxppRep-Setbaseyqq∪ppRep-Setbasexqqyy hproofi

1.17.21. Denition: Intersection

denition OclIntersection :: [(⁰A,⁰α::null)Set,(⁰A,⁰α)Set]⇒(⁰A,⁰α)Set where OclIntersection x y = (λ τ . if (δx) τ =true τ ∧(δ y)τ =true τ

then Abs-SetbasexxppRep-Setbase (y τ)qq

∩ppRep-Setbase(x τ)qqyy else ⊥ )

notation OclIntersection(-−>intersectionS et0(-⁰) )

lemma OclIntersection-inv: (x::Set(⁰b::{null}))6=⊥=⇒x 6=null =⇒ y 6=⊥ =⇒y 6=null =⇒ xxppRep-Setbase yqq∩ppRep-Setbase xqqyy∈ {X.X =bot ∨X =null ∨(∀x∈ppXqq.x 6=bot)}

hproofi

interpretation OclIntersection : prole-bind-d OclIntersection λx y. Abs-SetbasexxppRep-Setbase yqq ∩ ppRep-Setbasexqqyy

hproofi

1.17.22. Denition (future operators)

consts

OclCount :: [(⁰A,⁰α::null)Set,(⁰A,⁰α) Set]⇒ ⁰AInteger OclSum :: (⁰A,⁰α::null) Set⇒ ⁰AInteger

notation OclCount (-−>countS et0(-⁰) ) notation OclSum (-−>sumS et0

(⁰) )

1.17.23. Logical Properties

OclIncluding

lemma OclIncluding-valid-args-valid:

(τ |=υ(X−>includingS et(x))) = ((τ |=(δ X))∧(τ |=(υ x))) hproofi

lemma OclIncluding-valid-args-valid⁰⁰[simp,code-unfold]:

υ(X−>includingS et(x)) = ((δ X) and (υ x))

hproofi etc. etc.

OclExcluding

lemma OclExcluding-valid-args-valid:

(τ |=υ(X−>excludingS et(x))) = ((τ |=(δX))∧(τ |=(υ x))) hproofi

lemma OclExcluding-valid-args-valid⁰⁰[simp,code-unfold]:

υ(X−>excludingS et(x)) = ((δX)and (υ x)) hproofi

OclIncludes

lemma OclIncludes-valid-args-valid:

(τ |=υ(X−>includesS et(x))) = ((τ |=(δ X))∧(τ |=(υ x))) hproofi

lemma OclIncludes-valid-args-valid⁰⁰[simp,code-unfold]:

υ(X−>includesS et(x)) = ((δX)and (υ x)) hproofi

OclExcludes

lemma OclExcludes-valid-args-valid:

(τ |=υ(X−>excludesS et(x))) = ((τ |=(δ X))∧(τ |=(υx))) hproofi

lemma OclExcludes-valid-args-valid⁰⁰[simp,code-unfold]:

υ(X−>excludesS et(x)) = ((δ X) and (υ x)) hproofi

OclSize

lemma OclSize-dened-args-valid:τ |=δ(X−>sizeS et()) =⇒τ |=δ X hproofi

lemma OclSize-innite:

assumes non-nite:τ |=not(δ(S−>sizeS et()))

shows (τ |=not(δ(S)))∨ ¬ niteppRep-Setbase (S τ)qq hproofi

lemmaτ |=δ X =⇒ ¬nite ppRep-Setbase (X τ)qq=⇒ ¬τ |=δ(X−>sizeS et()) hproofi

lemma size-dened: assumes X-nite:V

τ . niteppRep-Setbase(X τ)qq shows δ(X−>sizeS et()) =δ X

hproofi

lemma size-dened⁰:

assumes X-nite:nite ppRep-Setbase(X τ)qq shows (τ |=δ(X−>sizeS et())) = (τ |=δ X) hproofi

OclIsEmpty

lemma OclIsEmpty-dened-args-valid:τ |=δ(X−>isEmptyS et()) =⇒τ |=υ X hproofi

lemmaτ |=δ(null−>isEmptyS et()) hproofi

lemma OclIsEmpty-innite:τ |=δ X =⇒ ¬niteppRep-Setbase (X τ)qq=⇒ ¬τ |=δ (X−>isEmptyS et()) hproofi

OclNotEmpty

lemma OclNotEmpty-dened-args-valid:τ |=δ(X−>notEmptyS et()) =⇒τ |=υ X hproofi

lemmaτ |=δ(null−>notEmptyS et()) hproofi

lemma OclNotEmpty-innite:τ |=δX =⇒ ¬niteppRep-Setbase(X τ)qq=⇒ ¬τ |=δ(X−>notEmptyS et()) hproofi

lemma OclNotEmpty-has-elt :τ |=δ X =⇒ τ |=X−>notEmptyS et() =⇒

∃e.e ∈ppRep-Setbase(X τ)qq hproofi

OclANY

lemma OclANY-dened-args-valid:τ |=δ(X−>anyS et()) =⇒τ |=δ X hproofi

lemmaτ |=δX =⇒τ |=X−>isEmptyS et() =⇒ ¬τ |=δ (X−>anyS et()) hproofi

lemma OclANY-valid-args-valid: (τ |=υ(X−>anyS et())) = (τ |=υ X) hproofi

lemma OclANY-valid-args-valid⁰⁰[simp,code-unfold]:

υ(X−>anyS et()) = (υ X) hproofi

Dans le document FEATHERWEIGHT OCL (Page 91-97)