Package h-def: Definition of memory safety for the H API

Information

nameh-def
version1.92
descriptionDefinition of memory safety for the H API
authorJoe Hurd <joe@gilith.com>
licenseMIT
provenanceHOL Light theory extracted on 2012-08-13
requiresbase
showData.Bool
Data.Byte
Data.List
Data.Option
Data.Pair
Data.Word10
Data.Word12
Function
Number.Natural
Set
System.H

Files

Defined Type Operators

Defined Constants

Theorems

⊦ View.mkCurrentPageDirectoryH = State.cr3

⊦ View.mkReferencePageDirectoryH = State.reference

⊦ View.mkRegionHandlesH = State.regions

⊦ View.mkRegionHandlesKernel = State.regions

⊦ Page.Data.zero = Page.Data.mk (λi. 0)

⊦ ∀f. View.observablePagesEnvironment (View.Environment f) = f

⊦ ∀f. View.observablePagesUser (View.User f) = f

⊦ ∀n.
    Physical.Superpage.fromNatural n =
    Physical.Superpage.mk (fromNatural n)

⊦ ∀n. Superpage.Offset.fromNatural n = Superpage.Offset.mk (fromNatural n)

⊦ ∀n.
    Virtual.Superpage.fromNatural n = Virtual.Superpage.mk (fromNatural n)

⊦ ∀p. Page.isDirectory p ⇔ isSome (Page.destDirectory p)

⊦ ∀p. Page.isEnvironment p ⇔ isSome (Page.destEnvironment p)

⊦ ∀p. Page.isInstalled p ⇔ ¬Page.isNotInstalled p

⊦ ∀p. Page.isNormal p ⇔ isSome (Page.destNormal p)

⊦ ∀p. Page.isTable p ⇔ isSome (Page.destTable p)

⊦ ∀va. Virtual.Address.isUser va ⇔ ¬Virtual.Address.isKernel va

⊦ ∀vpa. Virtual.Page.isUser vpa ⇔ ¬Virtual.Page.isKernel vpa

⊦ ∀vr.
    Virtual.Region.isKernel vr ⇔
    Virtual.Region.forall Virtual.Page.isKernel vr

⊦ ∀vr.
    Virtual.Region.isUser vr ⇔ Virtual.Region.forall Virtual.Page.isUser vr

⊦ ∀vsa. Virtual.Superpage.isUser vsa ⇔ ¬Virtual.Superpage.isKernel vsa

⊦ ∀s l. Physical.Region.length (Physical.Region.mk s l) = l

⊦ ∀s l. Physical.Region.start (Physical.Region.mk s l) = s

⊦ ∀f g. View.regionHandlesKernel (View.Kernel f g) = g

⊦ ∀f g. View.observablePagesKernel (View.Kernel f g) = f

⊦ ∀s l. Virtual.Region.length (Virtual.Region.mk s l) = l

⊦ ∀s l. Virtual.Region.start (Virtual.Region.mk s l) = s

⊦ ∀i a. State.Regions.all (State.Regions i a) = a

⊦ ∀i a. State.Regions.initial (State.Regions i a) = i

⊦ View.mkObservablePagesKernel =
  View.ObservablePages.fromTranslate
    (λs vpa. State.translatePage s (State.cr3 s) vpa)

⊦ ∀p. Page.isDirectoryOrTable p ⇔ Page.isDirectory p ∨ Page.isTable p

⊦ ∀s.
    State.cr3IsPageDirectory s ⇔
    Page.isDirectory (State.page s (State.cr3 s))

⊦ ∀s.
    State.referenceIsPageDirectory s ⇔
    Page.isDirectory (State.page s (State.reference s))

⊦ ∀s ppa.
    State.isEnvironmentPage s ppa ⇔ Page.isEnvironment (State.page s ppa)

⊦ ∀s ppa. State.isNormalPage s ppa ⇔ Page.isNormal (State.page s ppa)

⊦ ∀ppa pr.
    Physical.Region.member ppa pr ⇔
    member ppa (Physical.Region.pageList pr)

⊦ ∀ppa vr.
    Virtual.Region.member ppa vr ⇔ member ppa (Virtual.Region.pageList vr)

⊦ ∀p pr.
    Physical.Region.exists p pr ⇔ exists p (Physical.Region.pageList pr)

⊦ ∀p pr. Physical.Region.forall p pr ⇔ all p (Physical.Region.pageList pr)

⊦ ∀p vr. Virtual.Region.exists p vr ⇔ exists p (Virtual.Region.pageList vr)

⊦ ∀p vr. Virtual.Region.forall p vr ⇔ all p (Virtual.Region.pageList vr)

⊦ ∀s.
    State.initialRegionsAreRegions s ⇔
    State.Regions.initial (State.regions s) ⊆
    State.Regions.all (State.regions s)

⊦ ∀a s. output a s = Output.mk (view (domain a) s)

⊦ ∀pr1 pr2.
    Physical.Region.isSubregion pr1 pr2 ⇔
    Physical.Region.forall (λppa. Physical.Region.member ppa pr2) pr1

⊦ ∀vr1 vr2.
    Virtual.Region.isSubregion vr1 vr2 ⇔
    Virtual.Region.forall (λppa. Virtual.Region.member ppa vr2) vr1

⊦ ∀psa n.
    Physical.Superpage.add psa n =
    Physical.Superpage.mk (Physical.Superpage.dest psa + fromNatural n)

⊦ ∀si n.
    Superpage.Offset.add si n =
    Superpage.Offset.mk (Superpage.Offset.dest si + fromNatural n)

⊦ ∀vsa n.
    Virtual.Superpage.add vsa n =
    Virtual.Superpage.mk (Virtual.Superpage.dest vsa + fromNatural n)

⊦ ∀p. (∀s l. p (Physical.Region.mk s l)) ⇒ ∀pr. p pr

⊦ ∀P. (∀a0 a1. P (State.Regions a0 a1)) ⇒ ∀x. P x

⊦ ∀p. (∀s l. p (Virtual.Region.mk s l)) ⇒ ∀pr. p pr

⊦ View.mkObservablePagesUser =
  View.ObservablePages.fromTranslate
    (λs vpa.
       if Virtual.Page.isUser vpa then
         State.translatePage s (State.cr3 s) vpa
       else none)

⊦ ∀s ppa.
    State.unmappedNormalPage s ppa ⇔
    State.unmappedPage s ppa ∧ State.isNormalPage s ppa

⊦ ∀s.
    State.referenceMapsKernelAddresses s ⇔
    ∀vpa.
      State.mappedPage s (State.reference s) vpa ⇒
      Virtual.Page.isKernel vpa

⊦ ∀c r s g. State.cr3 (State c r s g) = c

⊦ ∀c r s g. State.reference (State c r s g) = r

⊦ ∀c r s g. State.regions (State c r s g) = g

⊦ ∀c r s g. State.page (State c r s g) = s

⊦ ∀c p g r. View.currentPageDirectoryH (View.H c p g r) = c

⊦ ∀c p g r. View.referencePageDirectoryH (View.H c p g r) = r

⊦ ∀c p g r. View.regionHandlesH (View.H c p g r) = g

⊦ ∀c p g r. View.pagesH (View.H c p g r) = p

⊦ ∀f. ∃fn. ∀s l. fn (Physical.Region.mk s l) = f s l

⊦ ∀f. ∃fn. ∀s l. fn (Virtual.Region.mk s l) = f s l

⊦ ∀f. ∃fn. ∀a0 a1. fn (State.Regions a0 a1) = f a0 a1

⊦ (∀x. Output.mk (Output.dest x) = x) ∧ ∀v. Output.dest (Output.mk v) = v

⊦ (∀l. RegionLength.fromNatural (RegionLength.toNatural l) = l) ∧
  ∀n. RegionLength.toNatural (RegionLength.fromNatural n) = n

⊦ (∀d. Page.Data.mk (Page.Data.dest d) = d) ∧
  ∀f. Page.Data.dest (Page.Data.mk f) = f

⊦ (∀i. Page.Offset.mk (Page.Offset.dest i) = i) ∧
  ∀w. Page.Offset.dest (Page.Offset.mk w) = w

⊦ (∀d. Page.Directory.Data.mk (Page.Directory.Data.dest d) = d) ∧
  ∀f. Page.Directory.Data.dest (Page.Directory.Data.mk f) = f

⊦ (∀t. Page.Table.Data.mk (Page.Table.Data.dest t) = t) ∧
  ∀f. Page.Table.Data.dest (Page.Table.Data.mk f) = f

⊦ (∀pa. Physical.Address.mk (Physical.Address.dest pa) = pa) ∧
  ∀r. Physical.Address.dest (Physical.Address.mk r) = r

⊦ (∀ppa. Physical.Page.mk (Physical.Page.dest ppa) = ppa) ∧
  ∀r. Physical.Page.dest (Physical.Page.mk r) = r

⊦ (∀psa. Physical.Superpage.mk (Physical.Superpage.dest psa) = psa) ∧
  ∀w. Physical.Superpage.dest (Physical.Superpage.mk w) = w

⊦ (∀si. Superpage.Offset.mk (Superpage.Offset.dest si) = si) ∧
  ∀w. Superpage.Offset.dest (Superpage.Offset.mk w) = w

⊦ (∀v. View.ObservablePages.mk (View.ObservablePages.dest v) = v) ∧
  ∀f. View.ObservablePages.dest (View.ObservablePages.mk f) = f

⊦ (∀va. Virtual.Address.mk (Virtual.Address.dest va) = va) ∧
  ∀r. Virtual.Address.dest (Virtual.Address.mk r) = r

⊦ (∀vpa. Virtual.Page.mk (Virtual.Page.dest vpa) = vpa) ∧
  ∀r. Virtual.Page.dest (Virtual.Page.mk r) = r

⊦ (∀vsa. Virtual.Superpage.mk (Virtual.Superpage.dest vsa) = vsa) ∧
  ∀w. Virtual.Superpage.dest (Virtual.Superpage.mk w) = w

⊦ ∀s pd vpa.
    State.mappedPage s pd vpa ⇔ isSome (State.translatePage s pd vpa)

⊦ ∀s l.
    Physical.Region.pageList (Physical.Region.mk s l) =
    map (Physical.Page.add s) (interval 0 (RegionLength.toNatural l))

⊦ ∀s l.
    Virtual.Region.pageList (Virtual.Region.mk s l) =
    map (Virtual.Page.add s) (interval 0 (RegionLength.toNatural l))

⊦ ∀P.
    (∀a. P (Page.Directory.Entry.Superpage a)) ∧
    (∀a. P (Page.Directory.Entry.Table a)) ⇒ ∀x. P x

⊦ ∀d s t. View.equivalent d s t ⇔ view d s = view d t

⊦ ∀s ppa.
    View.mkPagesH s ppa =
    let page ← State.page s ppa in
    if Page.isNormal page then none else some page

⊦ ∀p.
    p Domain.Environment ∧ p Domain.H ∧ p Domain.Kernel ∧ p Domain.User ⇒
    ∀d. p d

⊦ ∀s.
    State.mappedPagesAreAvailable s ⇔
    ∀pd vpa.
      case State.translatePage s pd vpa of
        none → ⊤
      | some ppa → Page.isInstalled (State.page s ppa)

⊦ ∀va.
    Virtual.Address.isKernel va ⇔
    let (vpa, i) ← Virtual.Address.dest va in Virtual.Page.isKernel vpa

⊦ ∀vpa.
    Virtual.Page.isKernel vpa ⇔
    let (vsa, si) ← Virtual.Page.dest vpa in Virtual.Superpage.isKernel vsa

⊦ ∀P. (∀a0 a1 a2 a3. P (State a0 a1 a2 a3)) ⇒ ∀x. P x

⊦ ∀k v d.
    Page.Data.update k v d =
    Page.Data.mk (λi. if i = k then v else Page.Data.dest d i)

⊦ ∀a s s'.
    action a s s' ⇔
    State.wellformed s ∧ State.wellformed s' ∧ Action.specification a s s'

⊦ View.mkObservablePagesEnvironment =
  View.ObservablePages.fromTranslate
    (λs vpa.
       case State.translatePage s (State.cr3 s) vpa of
         none → none
       | some ppa →
           if Page.isEnvironment (State.page s ppa) then some ppa
           else none)

⊦ (∀ppa. Physical.Page.add ppa 0 = ppa) ∧
  ∀ppa n.
    Physical.Page.add ppa (suc n) =
    Physical.Page.add (Physical.Page.suc ppa) n

⊦ (∀vpa. Virtual.Page.add vpa 0 = vpa) ∧
  ∀vpa n.
    Virtual.Page.add vpa (suc n) =
    Virtual.Page.add (Virtual.Page.suc vpa) n

⊦ ∀s.
    State.referenceContainsEnvironment s ⇔
    ∀ppa.
      Page.isEnvironment (State.page s ppa) ⇒
      ∃vpa. State.translatePage s (State.reference s) vpa = some ppa

⊦ ∀pd s s'.
    Action.specificationFreePageDirectoryH pd s s' ⇔
    Action.specificationAllocatePageDirectoryH pd s' s ∧
    State.page s' pd = Page.Normal Page.Data.zero

⊦ ∀f0 f1.
    ∃fn.
      (∀a. fn (Page.Directory.Entry.Superpage a) = f0 a) ∧
      ∀a. fn (Page.Directory.Entry.Table a) = f1 a

⊦ ∀f. ∃fn. ∀a0 a1 a2 a3. fn (State a0 a1 a2 a3) = f a0 a1 a2 a3

⊦ ∀vsa.
    Virtual.Superpage.isKernel vsa ⇔
    let w ← Virtual.Superpage.dest vsa in bit w 9 ∧ bit w 8

⊦ ∀s.
    State.regionsAreNotEnvironment s ⇔
    ∀r ppa.
      r ∈ State.Regions.all (State.regions s) ∧
      Physical.Region.member ppa r ⇒ ¬Page.isEnvironment (State.page s ppa)

⊦ ∀s ppa.
    State.unmappedPage s ppa ⇔
    ∀pd.
      Page.isDirectory (State.page s pd) ⇒
      ∀vpa. ¬(State.translatePage s pd vpa = some ppa)

⊦ ∀s.
    State.environmentOnlyInReference s ⇔
    ∀pd vpa.
      case State.translatePage s pd vpa of
        none → ⊤
      | some ppa →
          Page.isEnvironment (State.page s ppa) ⇒
          State.mappedPage s (State.reference s) vpa

⊦ ∀s pd pt.
    State.tableMappedInDirectory s pd pt ⇔
    case Page.destDirectory (State.page s pd) of
      none → ⊥
    | some pdd →
        ∃pdi.
          Page.Directory.Data.dest pdd pdi =
          some (Page.Directory.Entry.Table pt)

⊦ ∀s.
    State.pageDirectoriesContainReference s ⇔
    ∀pd vpa.
      Page.isDirectory (State.page s pd) ∧ Virtual.Page.isKernel vpa ⇒
      State.translatePage s pd vpa =
      State.translatePage s (State.reference s) vpa

⊦ (∀f g psa.
     Page.Directory.Entry.case f g (Page.Directory.Entry.Superpage psa) =
     f psa) ∧
  ∀f g pt.
    Page.Directory.Entry.case f g (Page.Directory.Entry.Table pt) = g pt

⊦ ∀fe fh fk fu.
    ∃fn.
      fn Domain.Environment = fe ∧ fn Domain.H = fh ∧
      fn Domain.Kernel = fk ∧ fn Domain.User = fu

⊦ ∀s.
    State.tablePointersArePageTables s ⇔
    ∀pd pdi.
      case Page.destDirectory (State.page s pd) of
        none → ⊤
      | some pdd →
          case Page.Directory.Data.dest pdd pdi of
            none → ⊤
          | some pde →
              case pde of
                Page.Directory.Entry.Superpage spa → ⊤
              | Page.Directory.Entry.Table ppa →
                  Page.isTable (State.page s ppa)

⊦ ∀P.
    (∀a. P (Page.Environment a)) ∧ (∀a. P (Page.Normal a)) ∧
    (∀a. P (Page.Directory a)) ∧ (∀a. P (Page.Table a)) ∧
    P Page.NotInstalled ⇒ ∀x. P x

⊦ ∀s va.
    State.translation s va =
    let (vpa, i) ← Virtual.Address.dest va in
    case State.translatePage s (State.cr3 s) vpa of
      none → none
    | some ppa → some (Physical.Address.mk (ppa, i))

⊦ ∀pd s s'.
    Action.specificationExecuteH pd s s' ⇔
    State.reference s = State.reference s' ∧ State.page s = State.page s' ∧
    State.regions s = State.regions s' ∧ State.cr3 s' = pd

⊦ ∀s.
    State.wellformed s ⇔
    State.cr3IsPageDirectory s ∧ State.pageDirectoriesContainReference s ∧
    State.mappedPagesAreAvailable s ∧ State.tablePointersArePageTables s ∧
    State.noSharedPageTables s ∧ State.referenceIsPageDirectory s ∧
    State.referenceMapsKernelAddresses s ∧
    State.environmentOnlyInReference s ∧ State.initialRegionsAreRegions s ∧
    State.regionsAreNotEnvironment s

⊦ (∀d. Page.isNotInstalled (Page.Environment d) ⇔ ⊥) ∧
  (∀d. Page.isNotInstalled (Page.Normal d) ⇔ ⊥) ∧
  (∀pdd. Page.isNotInstalled (Page.Directory pdd) ⇔ ⊥) ∧
  (∀ptd. Page.isNotInstalled (Page.Table ptd) ⇔ ⊥) ∧
  (Page.isNotInstalled Page.NotInstalled ⇔ ⊤)

⊦ ∀ppa.
    Physical.Page.suc ppa =
    let (psa, si) ← Physical.Page.dest ppa in
    let si' ← Superpage.Offset.add si 1 in
    let psa' ← if si' = 0 then Physical.Superpage.add psa 1 else psa in
    Physical.Page.mk (psa', si')

⊦ ∀vpa.
    Virtual.Page.suc vpa =
    let (vsa, si) ← Virtual.Page.dest vpa in
    let si' ← Superpage.Offset.add si 1 in
    let vsa' ← if si' = 0 then Virtual.Superpage.add vsa 1 else vsa in
    Virtual.Page.mk (vsa', si')

⊦ (∀d. Page.destNormal (Page.Environment d) = none) ∧
  (∀d. Page.destNormal (Page.Normal d) = some d) ∧
  (∀pdd. Page.destNormal (Page.Directory pdd) = none) ∧
  (∀ptd. Page.destNormal (Page.Table ptd) = none) ∧
  Page.destNormal Page.NotInstalled = none

⊦ (∀d. Page.destDirectory (Page.Environment d) = none) ∧
  (∀d. Page.destDirectory (Page.Normal d) = none) ∧
  (∀pdd. Page.destDirectory (Page.Directory pdd) = some pdd) ∧
  (∀ptd. Page.destDirectory (Page.Table ptd) = none) ∧
  Page.destDirectory Page.NotInstalled = none

⊦ (∀d. Page.destTable (Page.Environment d) = none) ∧
  (∀d. Page.destTable (Page.Normal d) = none) ∧
  (∀pdd. Page.destTable (Page.Directory pdd) = none) ∧
  (∀ptd. Page.destTable (Page.Table ptd) = some ptd) ∧
  Page.destTable Page.NotInstalled = none

⊦ (∀d. Page.destEnvironment (Page.Environment d) = some d) ∧
  (∀d. Page.destEnvironment (Page.Normal d) = none) ∧
  (∀pdd. Page.destEnvironment (Page.Directory pdd) = none) ∧
  (∀ptd. Page.destEnvironment (Page.Table ptd) = none) ∧
  Page.destEnvironment Page.NotInstalled = none

⊦ (∀d. Page.destEnvironmentOrNormal (Page.Environment d) = some d) ∧
  (∀d. Page.destEnvironmentOrNormal (Page.Normal d) = some d) ∧
  (∀pdd. Page.destEnvironmentOrNormal (Page.Directory pdd) = none) ∧
  (∀ptd. Page.destEnvironmentOrNormal (Page.Table ptd) = none) ∧
  Page.destEnvironmentOrNormal Page.NotInstalled = none

⊦ ∀tr s.
    View.ObservablePages.fromTranslate tr s =
    View.ObservablePages.mk
      (λvpa.
         case tr s vpa of
           none → none
         | some ppa →
             case Page.destEnvironmentOrNormal (State.page s ppa) of
               none → none
             | some d → some (d, { vpa'. vpa' | tr s vpa' = some ppa }))

⊦ ∀phi.
    (∀f. phi (View.Environment f)) ∧ (∀c p g r. phi (View.H c p g r)) ∧
    (∀f g. phi (View.Kernel f g)) ∧ (∀f. phi (View.User f)) ⇒ ∀v. phi v

⊦ (∀s.
     view Domain.Environment s =
     View.Environment (View.mkObservablePagesEnvironment s)) ∧
  (∀s.
     view Domain.H s =
     View.H (View.mkCurrentPageDirectoryH s) (View.mkPagesH s)
       (View.mkRegionHandlesH s) (View.mkReferencePageDirectoryH s)) ∧
  (∀s.
     view Domain.Kernel s =
     View.Kernel (View.mkObservablePagesKernel s)
       (View.mkRegionHandlesKernel s)) ∧
  ∀s. view Domain.User s = View.User (View.mkObservablePagesUser s)

⊦ ∀ppa s s'.
    Action.specificationAllocatePageDirectoryH ppa s s' ⇔
    State.cr3 s = State.cr3 s' ∧ State.reference s = State.reference s' ∧
    State.regions s = State.regions s' ∧ State.unmappedNormalPage s ppa ∧
    ∀ppa'.
      State.page s' ppa' =
      State.page s (if ppa' = ppa then State.reference s else ppa')

⊦ ∀pd pts pr vr s s'.
    Action.specificationAddMappingH pd pts pr vr s s' ⇔
    length (nub pts) = length pts ∧
    pr ∈ State.Regions.all (State.regions s) ∧
    ∃n.
      n ≤ length pts ∧
      Action.specificationAddMapping pd (toSet (take n pts)) pr vr s s'

⊦ ∀f0 f1 f2 f3 f4.
    ∃fn.
      (∀a. fn (Page.Environment a) = f0 a) ∧
      (∀a. fn (Page.Normal a) = f1 a) ∧
      (∀a. fn (Page.Directory a) = f2 a) ∧ (∀a. fn (Page.Table a) = f3 a) ∧
      fn Page.NotInstalled = f4

⊦ ∀pd vr s s'.
    Action.specificationRemoveMappingH pd vr s s' ⇔
    case State.translatePage s pd (Virtual.Region.start vr) of
      none → ⊥
    | some prs →
        let pr ← Physical.Region.mk prs (Virtual.Region.length vr) in
        let pts ←
            { ppa. ppa |
              Page.isTable (State.page s ppa) ∧
              State.page s' ppa = Page.Normal Page.Data.zero } in
        Action.specificationAddMapping pd pts pr vr s' s

⊦ ∀s pd vpa.
    State.translatePage s pd vpa =
    let (vsa, si) ← Virtual.Page.dest vpa in
    case Page.destDirectory (State.page s pd) of
      none → none
    | some pdd →
        case Page.Directory.Data.dest pdd vsa of
          none → none
        | some pde →
            case pde of
              Page.Directory.Entry.Superpage psa →
                some (Physical.Page.mk (psa, si))
            | Page.Directory.Entry.Table pt →
                case Page.destTable (State.page s pt) of
                  none → none
                | some ptd → Page.Table.Data.dest ptd si

⊦ ∀fe fh fk fu.
    ∃fn.
      (∀f. fn (View.Environment f) = fe f) ∧
      (∀c p g r. fn (View.H c p g r) = fh c p g r) ∧
      (∀f g. fn (View.Kernel f g) = fk f g) ∧ ∀f. fn (View.User f) = fu f

⊦ ∀s.
    State.noSharedPageTables s ⇔
    ∀pt pd1 pd2 pdd1 pdd2 vsa1 vsa2.
      Page.destDirectory (State.page s pd1) = some pdd1 ∧
      Page.destDirectory (State.page s pd2) = some pdd2 ∧
      Page.Directory.Data.dest pdd1 vsa1 =
      some (Page.Directory.Entry.Table pt) ∧
      Page.Directory.Data.dest pdd2 vsa2 =
      some (Page.Directory.Entry.Table pt) ⇒
      vsa1 = vsa2 ∧ (Virtual.Superpage.isKernel vsa1 ∨ pd1 = pd2)

⊦ ∀pr ppa l s s'.
    Action.specificationDeriveRegionH pr ppa l s s' ⇔
    State.cr3 s = State.cr3 s' ∧ State.reference s = State.reference s' ∧
    State.page s = State.page s' ∧
    State.Regions.initial (State.regions s) =
    State.Regions.initial (State.regions s') ∧
    let r ← Physical.Region.mk ppa l in
    Physical.Region.isSubregion r pr ∧
    ¬(r ∈ State.Regions.all (State.regions s)) ∧
    State.Regions.all (State.regions s') =
    insert r (State.Regions.all (State.regions s))

⊦ ∀va b s s'.
    Action.specificationWriteEnvironment va b s s' ⇔
    State.cr3 s = State.cr3 s' ∧ State.reference s = State.reference s' ∧
    State.regions s = State.regions s' ∧
    case State.translation s va of
      none → ⊥
    | some pa →
        let (ppa, i) ← Physical.Address.dest pa in
        ∀ppa'.
          if ppa' = ppa then
            case Page.destEnvironment (State.page s ppa') of
              none → ⊥
            | some d →
                Page.destEnvironment (State.page s' ppa') =
                some (Page.Data.update i b d)
          else State.page s ppa' = State.page s' ppa'

⊦ ∀va b s s'.
    Action.specificationWriteKernel va b s s' ⇔
    State.cr3 s = State.cr3 s' ∧ State.reference s = State.reference s' ∧
    State.regions s = State.regions s' ∧ Virtual.Address.isKernel va ∧
    case State.translation s va of
      none → ⊥
    | some pa →
        let (ppa, i) ← Physical.Address.dest pa in
        ∀ppa'.
          if ppa' = ppa then
            case Page.destNormal (State.page s ppa') of
              none → ⊥
            | some d →
                Page.destNormal (State.page s' ppa') =
                some (Page.Data.update i b d)
          else State.page s ppa' = State.page s' ppa'

⊦ ∀va b s s'.
    Action.specificationWriteUser va b s s' ⇔
    State.cr3 s = State.cr3 s' ∧ State.reference s = State.reference s' ∧
    State.regions s = State.regions s' ∧ Virtual.Address.isUser va ∧
    case State.translation s va of
      none → ⊥
    | some pa →
        let (ppa, i) ← Physical.Address.dest pa in
        ∀ppa'.
          if ppa' = ppa then
            case Page.destNormal (State.page s ppa') of
              none → ⊥
            | some d →
                Page.destNormal (State.page s' ppa') =
                some (Page.Data.update i b d)
          else State.page s ppa' = State.page s' ppa'

⊦ ∀p.
    (∀va b. p (Action.WriteEnvironment va b)) ∧
    (∀pr ppa l. p (Action.DeriveRegionH pr ppa l)) ∧
    (∀ppa. p (Action.AllocatePageDirectoryH ppa)) ∧
    (∀pd. p (Action.FreePageDirectoryH pd)) ∧
    (∀pd pts pr vr. p (Action.AddMappingH pd pts pr vr)) ∧
    (∀pd vr. p (Action.RemoveMappingH pd vr)) ∧
    (∀pr vr. p (Action.AddKernelMappingH pr vr)) ∧
    (∀pd. p (Action.ExecuteH pd)) ∧ (∀va b. p (Action.WriteKernel va b)) ∧
    (∀va b. p (Action.WriteUser va b)) ⇒ ∀a. p a

⊦ ∀x y.
    interferes x y ⇔
    x = Domain.Environment ∧ y = Domain.Environment ∨
    x = Domain.Environment ∧ y = Domain.H ∨
    x = Domain.Environment ∧ y = Domain.Kernel ∨
    x = Domain.H ∧ y = Domain.H ∨ x = Domain.H ∧ y = Domain.Kernel ∨
    x = Domain.H ∧ y = Domain.User ∨
    x = Domain.Kernel ∧ y = Domain.Kernel ∨
    x = Domain.Kernel ∧ y = Domain.User ∨
    x = Domain.User ∧ y = Domain.Kernel ∨ x = Domain.User ∧ y = Domain.User

⊦ (∀va b. domain (Action.WriteEnvironment va b) = Domain.Environment) ∧
  (∀pr ppa l. domain (Action.DeriveRegionH pr ppa l) = Domain.H) ∧
  (∀ppa. domain (Action.AllocatePageDirectoryH ppa) = Domain.H) ∧
  (∀pd. domain (Action.FreePageDirectoryH pd) = Domain.H) ∧
  (∀pd pts pr vr. domain (Action.AddMappingH pd pts pr vr) = Domain.H) ∧
  (∀pd vr. domain (Action.RemoveMappingH pd vr) = Domain.H) ∧
  (∀pr vr. domain (Action.AddKernelMappingH pr vr) = Domain.H) ∧
  (∀pd. domain (Action.ExecuteH pd) = Domain.H) ∧
  (∀va b. domain (Action.WriteKernel va b) = Domain.Kernel) ∧
  ∀va b. domain (Action.WriteUser va b) = Domain.User

⊦ ∀pr vr s s'.
    Action.specificationAddKernelMappingH pr vr s s' ⇔
    State.cr3 s = State.cr3 s' ∧ State.reference s = State.reference s' ∧
    State.regions s = State.regions s' ∧
    pr ∈ State.Regions.all (State.regions s) ∧
    Physical.Region.forall (State.isNormalPage s) pr ∧
    Physical.Region.length pr = Virtual.Region.length vr ∧
    Virtual.Region.isKernel vr ∧
    (∀vpa.
       ¬Virtual.Region.member vpa vr ⇒
       State.translatePage s' (State.reference s') vpa =
       State.translatePage s (State.reference s) vpa) ∧
    Virtual.Region.forall
      (λvpa.
         case State.translatePage s (State.reference s) vpa of
           none → ⊤
         | some ppa → ¬State.isEnvironmentPage s ppa) vr ∧
    all id
      (zipWith
         (λvpa ppa.
            State.translatePage s' (State.reference s') vpa = some ppa)
         (Virtual.Region.pageList vr) (Physical.Region.pageList pr)) ∧
    ∀ppa.
      if State.tableMappedInDirectory s (State.reference s) ppa then
        Page.isTable (State.page s' ppa)
      else State.page s ppa = State.page s' ppa

⊦ (∀va b.
     Action.specification (Action.WriteEnvironment va b) =
     Action.specificationWriteEnvironment va b) ∧
  (∀pr ppa l.
     Action.specification (Action.DeriveRegionH pr ppa l) =
     Action.specificationDeriveRegionH pr ppa l) ∧
  (∀ppa.
     Action.specification (Action.AllocatePageDirectoryH ppa) =
     Action.specificationAllocatePageDirectoryH ppa) ∧
  (∀pd.
     Action.specification (Action.FreePageDirectoryH pd) =
     Action.specificationFreePageDirectoryH pd) ∧
  (∀pd pts pr vr.
     Action.specification (Action.AddMappingH pd pts pr vr) =
     Action.specificationAddMappingH pd pts pr vr) ∧
  (∀pd vr.
     Action.specification (Action.RemoveMappingH pd vr) =
     Action.specificationRemoveMappingH pd vr) ∧
  (∀pr vr.
     Action.specification (Action.AddKernelMappingH pr vr) =
     Action.specificationAddKernelMappingH pr vr) ∧
  (∀pd.
     Action.specification (Action.ExecuteH pd) =
     Action.specificationExecuteH pd) ∧
  (∀va b.
     Action.specification (Action.WriteKernel va b) =
     Action.specificationWriteKernel va b) ∧
  ∀va b.
    Action.specification (Action.WriteUser va b) =
    Action.specificationWriteUser va b

⊦ ∀fwe fdr fapd ffpd fam frm fakm fe fwk fwu.
    ∃fn.
      (∀va b. fn (Action.WriteEnvironment va b) = fwe va b) ∧
      (∀pr ppa l. fn (Action.DeriveRegionH pr ppa l) = fdr pr ppa l) ∧
      (∀ppa. fn (Action.AllocatePageDirectoryH ppa) = fapd ppa) ∧
      (∀pd. fn (Action.FreePageDirectoryH pd) = ffpd pd) ∧
      (∀pd pts pr vr.
         fn (Action.AddMappingH pd pts pr vr) = fam pd pts pr vr) ∧
      (∀pd vr. fn (Action.RemoveMappingH pd vr) = frm pd vr) ∧
      (∀pr vr. fn (Action.AddKernelMappingH pr vr) = fakm pr vr) ∧
      (∀pd. fn (Action.ExecuteH pd) = fe pd) ∧
      (∀va b. fn (Action.WriteKernel va b) = fwk va b) ∧
      ∀va b. fn (Action.WriteUser va b) = fwu va b

⊦ ∀pd pts pr vr s s'.
    Action.specificationAddMapping pd pts pr vr s s' ⇔
    State.cr3 s = State.cr3 s' ∧ State.reference s = State.reference s' ∧
    State.regions s = State.regions s' ∧
    Page.isDirectory (State.page s pd) ∧
    (∀pt. pt ∈ pts ⇒ State.unmappedNormalPage s pt) ∧
    (∀pt. pt ∈ pts ⇒ ¬Physical.Region.member pt pr) ∧
    Physical.Region.forall (State.isNormalPage s) pr ∧
    Physical.Region.length pr = Virtual.Region.length vr ∧
    Virtual.Region.isUser vr ∧
    (∀vpa.
       if Virtual.Region.member vpa vr then
         isNone (State.translatePage s pd vpa)
       else State.translatePage s' pd vpa = State.translatePage s pd vpa) ∧
    all id
      (zipWith (λvpa ppa. State.translatePage s' pd vpa = some ppa)
         (Virtual.Region.pageList vr) (Physical.Region.pageList pr)) ∧
    ∀ppa.
      if ppa ∈ pts then State.tableMappedInDirectory s' pd ppa
      else if ppa = pd then Page.isDirectory (State.page s' ppa)
      else if State.tableMappedInDirectory s pd ppa then
        Page.isTable (State.page s' ppa)
      else State.page s ppa = State.page s' ppa

Input Type Operators

Input Constants

Assumptions

⊦ ⊤

⊦ ¬⊥ ⇔ ⊤

⊦ ¬⊤ ⇔ ⊥

⊦ bit0 0 = 0

⊦ ∀n. 0 ≤ n

⊦ ⊥ ⇔ ∀p. p

⊦ (¬) = λp. p ⇒ ⊥

⊦ (∃) = λp. p ((select) p)

⊦ ∀a. ∃!x. x = a

⊦ ∀t. (∀x. t) ⇔ t

⊦ ∀t. (∃x. t) ⇔ t

⊦ ∀t. (λx. t x) = t

⊦ (∀) = λp. p = λx. ⊤

⊦ ∀t. ¬¬t ⇔ t

⊦ ∀t. (⊤ ⇔ t) ⇔ t

⊦ ∀t. (t ⇔ ⊤) ⇔ t

⊦ ∀t. ⊥ ∧ t ⇔ ⊥

⊦ ∀t. ⊤ ∧ t ⇔ t

⊦ ∀t. t ∧ ⊤ ⇔ t

⊦ ∀t. ⊥ ⇒ t ⇔ ⊤

⊦ ∀t. ⊤ ⇒ t ⇔ t

⊦ ∀t. t ⇒ ⊤ ⇔ ⊤

⊦ ∀t. ⊥ ∨ t ⇔ t

⊦ ∀t. ⊤ ∨ t ⇔ ⊤

⊦ ∀t. t ∨ ⊥ ⇔ t

⊦ ∀n. ¬(suc n = 0)

⊦ ∀n. 0 + n = n

⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t

⊦ ∀t. (t ⇔ ⊥) ⇔ ¬t

⊦ ∀t. t ⇒ ⊥ ⇔ ¬t

⊦ ∀n. even (2 * n)

⊦ ∀n. bit1 n = suc (bit0 n)

⊦ ∀m. m ↑ 0 = 1

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)

⊦ ∀n. even (suc n) ⇔ ¬even n

⊦ ∀m. m ≤ 0 ⇔ m = 0

⊦ ∀t1 t2. (if ⊥ then t1 else t2) = t2

⊦ ∀t1 t2. (if ⊤ then t1 else t2) = t1

⊦ ∀x y. fst (x, y) = x

⊦ ∀x y. snd (x, y) = y

⊦ ∀n. bit0 (suc n) = suc (suc (bit0 n))

⊦ ∀x y. x = y ⇔ y = x

⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1

⊦ ∀n. 2 * n = n + n

⊦ ∀m n. ¬(m < n ∧ n ≤ m)

⊦ ∀m n. ¬(m ≤ n ∧ n < m)

⊦ ∀m n. suc m ≤ n ⇔ m < n

⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤

⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q

⊦ ∀m n. m + suc n = suc (m + n)

⊦ ∀m n. suc m + n = suc (m + n)

⊦ ∀m n. suc m = suc n ⇔ m = n

⊦ ∀m n. even (m * n) ⇔ even m ∨ even n

⊦ ∀m n. even (m + n) ⇔ even m ⇔ even n

⊦ ∀m n. m ↑ suc n = m * m ↑ n

⊦ ∀f g. (∀x. f x = g x) ⇔ f = g

⊦ ∀p a. (∃x. a = x ∧ p x) ⇔ p a

⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r

⊦ ∀m n. m ≤ n ⇔ m < n ∨ m = n

⊦ ∀m n. m ≤ n ∧ n ≤ m ⇔ m = n

⊦ ∀p q. (∃x. p ∧ q x) ⇔ p ∧ ∃x. q x

⊦ ∀t1 t2 t3. (t1 ∧ t2) ∧ t3 ⇔ t1 ∧ t2 ∧ t3

⊦ ∀m n p. m * (n * p) = m * n * p

⊦ ∀m n p. m + p = n + p ⇔ m = n

⊦ ∀p x. (∀y. p y ⇔ y = x) ⇒ (select) p = x

⊦ ∀p. (∀x. ∃y. p x y) ⇔ ∃y. ∀x. p x (y x)

⊦ ∀m n. m ≤ suc n ⇔ m = suc n ∨ m ≤ n

⊦ ∀m n. m * n = 0 ⇔ m = 0 ∨ n = 0

⊦ ∀p. p 0 ∧ (∀n. p n ⇒ p (suc n)) ⇒ ∀n. p n

⊦ ∀m n. m ↑ n = 0 ⇔ m = 0 ∧ ¬(n = 0)

⊦ (∃!) = λp. (∃) p ∧ ∀x y. p x ∧ p y ⇒ x = y

⊦ ∀p q. (∀x. p x ∧ q x) ⇔ (∀x. p x) ∧ ∀x. q x

⊦ ∀p q. (∀x. p x ⇒ q x) ⇒ (∀x. p x) ⇒ ∀x. q x

⊦ ∀p q. (∀x. p x ⇒ q x) ⇒ (∃x. p x) ⇒ ∃x. q x

⊦ ∀p q. (∀x. p x) ∧ (∀x. q x) ⇔ ∀x. p x ∧ q x

⊦ ∀e f. ∃!fn. fn 0 = e ∧ ∀n. fn (suc n) = f (fn n) n

⊦ ∀m n p. m * n = m * p ⇔ m = 0 ∨ n = p

⊦ ∀m n p. m * n ≤ m * p ⇔ m = 0 ∨ n ≤ p

⊦ ∀m n p. m * n < m * p ⇔ ¬(m = 0) ∧ n < p

⊦ ∀p1 p2 q1 q2. (p1 ⇒ p2) ∧ (q1 ⇒ q2) ⇒ p1 ∧ q1 ⇒ p2 ∧ q2

⊦ ∀p1 p2 q1 q2. (p1 ⇒ p2) ∧ (q1 ⇒ q2) ⇒ p1 ∨ q1 ⇒ p2 ∨ q2

⊦ ∀p. (∀x. ∃!y. p x y) ⇔ ∃f. ∀x y. p x y ⇔ f x = y

⊦ ∀p c x y. p (if c then x else y) ⇔ (c ⇒ p x) ∧ (¬c ⇒ p y)

⊦ ∀p. (∃!x. p x) ⇔ (∃x. p x) ∧ ∀x x'. p x ∧ p x' ⇒ x = x'