Theory Alloc

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theory Alloc
imports AllocBase PPROD
uses [Alloc.ML]
begin

(*  Title:      HOL/UNITY/Alloc
    ID:         $Id: Alloc.thy,v 1.4 2005/09/07 19:00:30 wenzelm Exp $
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1998  University of Cambridge

Specification of Chandy and Charpentier's Allocator
*)

theory Alloc
imports AllocBase PPROD
begin

(** State definitions.  OUTPUT variables are locals **)

record clientState =
  giv :: "nat list"   (*client's INPUT history:  tokens GRANTED*)
  ask :: "nat list"   (*client's OUTPUT history: tokens REQUESTED*)
  rel :: "nat list"   (*client's OUTPUT history: tokens RELEASED*)

record 'a clientState_d =
  clientState +
  dummy :: 'a       (*dummy field for new variables*)

constdefs
  (*DUPLICATED FROM Client.thy, but with "tok" removed*)
  (*Maybe want a special theory section to declare such maps*)
  non_dummy :: "'a clientState_d => clientState"
    "non_dummy s == (|giv = giv s, ask = ask s, rel = rel s|)"

  (*Renaming map to put a Client into the standard form*)
  client_map :: "'a clientState_d => clientState*'a"
    "client_map == funPair non_dummy dummy"

  
record allocState =
  allocGiv :: "nat => nat list"   (*OUTPUT history: source of "giv" for i*)
  allocAsk :: "nat => nat list"   (*INPUT: allocator's copy of "ask" for i*)
  allocRel :: "nat => nat list"   (*INPUT: allocator's copy of "rel" for i*)

record 'a allocState_d =
  allocState +
  dummy    :: 'a                (*dummy field for new variables*)

record 'a systemState =
  allocState +
  client :: "nat => clientState"  (*states of all clients*)
  dummy  :: 'a                    (*dummy field for new variables*)


constdefs

(** Resource allocation system specification **)

  (*spec (1)*)
  system_safety :: "'a systemState program set"
    "system_safety ==
     Always {s. (SUM i: lessThan Nclients. (tokens o giv o sub i o client)s)
     <= NbT + (SUM i: lessThan Nclients. (tokens o rel o sub i o client)s)}"

  (*spec (2)*)
  system_progress :: "'a systemState program set"
    "system_progress == INT i : lessThan Nclients.
                        INT h. 
                          {s. h <= (ask o sub i o client)s} LeadsTo
                          {s. h pfixLe (giv o sub i o client) s}"

  system_spec :: "'a systemState program set"
    "system_spec == system_safety Int system_progress"

(** Client specification (required) ***)

  (*spec (3)*)
  client_increasing :: "'a clientState_d program set"
    "client_increasing ==
         UNIV guarantees  Increasing ask Int Increasing rel"

  (*spec (4)*)
  client_bounded :: "'a clientState_d program set"
    "client_bounded ==
         UNIV guarantees  Always {s. ALL elt : set (ask s). elt <= NbT}"

  (*spec (5)*)
  client_progress :: "'a clientState_d program set"
    "client_progress ==
         Increasing giv  guarantees
         (INT h. {s. h <= giv s & h pfixGe ask s}
                 LeadsTo {s. tokens h <= (tokens o rel) s})"

  (*spec: preserves part*)
  client_preserves :: "'a clientState_d program set"
    "client_preserves == preserves giv Int preserves clientState_d.dummy"

  (*environmental constraints*)
  client_allowed_acts :: "'a clientState_d program set"
    "client_allowed_acts ==
       {F. AllowedActs F =
            insert Id (UNION (preserves (funPair rel ask)) Acts)}"

  client_spec :: "'a clientState_d program set"
    "client_spec == client_increasing Int client_bounded Int client_progress
                    Int client_allowed_acts Int client_preserves"

(** Allocator specification (required) ***)

  (*spec (6)*)
  alloc_increasing :: "'a allocState_d program set"
    "alloc_increasing ==
         UNIV  guarantees
         (INT i : lessThan Nclients. Increasing (sub i o allocGiv))"

  (*spec (7)*)
  alloc_safety :: "'a allocState_d program set"
    "alloc_safety ==
         (INT i : lessThan Nclients. Increasing (sub i o allocRel))
         guarantees
         Always {s. (SUM i: lessThan Nclients. (tokens o sub i o allocGiv)s)
         <= NbT + (SUM i: lessThan Nclients. (tokens o sub i o allocRel)s)}"

  (*spec (8)*)
  alloc_progress :: "'a allocState_d program set"
    "alloc_progress ==
         (INT i : lessThan Nclients. Increasing (sub i o allocAsk) Int
                                     Increasing (sub i o allocRel))
         Int
         Always {s. ALL i<Nclients.
                     ALL elt : set ((sub i o allocAsk) s). elt <= NbT}
         Int
         (INT i : lessThan Nclients. 
          INT h. {s. h <= (sub i o allocGiv)s & h pfixGe (sub i o allocAsk)s}
                 LeadsTo
                 {s. tokens h <= (tokens o sub i o allocRel)s})
         guarantees
             (INT i : lessThan Nclients.
              INT h. {s. h <= (sub i o allocAsk) s}
                     LeadsTo
                     {s. h pfixLe (sub i o allocGiv) s})"

  (*NOTE: to follow the original paper, the formula above should have had
        INT h. {s. h i <= (sub i o allocGiv)s & h i pfixGe (sub i o allocAsk)s}
               LeadsTo
               {s. tokens h i <= (tokens o sub i o allocRel)s})
    thus h should have been a function variable.  However, only h i is ever
    looked at.*)

  (*spec: preserves part*)
  alloc_preserves :: "'a allocState_d program set"
    "alloc_preserves == preserves allocRel Int preserves allocAsk Int
                        preserves allocState_d.dummy"
  
  (*environmental constraints*)
  alloc_allowed_acts :: "'a allocState_d program set"
    "alloc_allowed_acts ==
       {F. AllowedActs F =
            insert Id (UNION (preserves allocGiv) Acts)}"

  alloc_spec :: "'a allocState_d program set"
    "alloc_spec == alloc_increasing Int alloc_safety Int alloc_progress Int
                   alloc_allowed_acts Int alloc_preserves"

(** Network specification ***)

  (*spec (9.1)*)
  network_ask :: "'a systemState program set"
    "network_ask == INT i : lessThan Nclients.
                        Increasing (ask o sub i o client)  guarantees
                        ((sub i o allocAsk) Fols (ask o sub i o client))"

  (*spec (9.2)*)
  network_giv :: "'a systemState program set"
    "network_giv == INT i : lessThan Nclients.
                        Increasing (sub i o allocGiv)
                        guarantees
                        ((giv o sub i o client) Fols (sub i o allocGiv))"

  (*spec (9.3)*)
  network_rel :: "'a systemState program set"
    "network_rel == INT i : lessThan Nclients.
                        Increasing (rel o sub i o client)
                        guarantees
                        ((sub i o allocRel) Fols (rel o sub i o client))"

  (*spec: preserves part*)
  network_preserves :: "'a systemState program set"
    "network_preserves ==
       preserves allocGiv  Int
       (INT i : lessThan Nclients. preserves (rel o sub i o client)  Int
                                   preserves (ask o sub i o client))"
  
  (*environmental constraints*)
  network_allowed_acts :: "'a systemState program set"
    "network_allowed_acts ==
       {F. AllowedActs F =
           insert Id
            (UNION (preserves allocRel Int
                    (INT i: lessThan Nclients. preserves(giv o sub i o client)))
                  Acts)}"

  network_spec :: "'a systemState program set"
    "network_spec == network_ask Int network_giv Int
                     network_rel Int network_allowed_acts Int
                     network_preserves"


(** State mappings **)
  sysOfAlloc :: "((nat => clientState) * 'a) allocState_d => 'a systemState"
    "sysOfAlloc == %s. let (cl,xtr) = allocState_d.dummy s
                       in (| allocGiv = allocGiv s,
                             allocAsk = allocAsk s,
                             allocRel = allocRel s,
                             client   = cl,
                             dummy    = xtr|)"


  sysOfClient :: "(nat => clientState) * 'a allocState_d => 'a systemState"
    "sysOfClient == %(cl,al). (| allocGiv = allocGiv al,
                                 allocAsk = allocAsk al,
                                 allocRel = allocRel al,
                                 client   = cl,
                                 systemState.dummy = allocState_d.dummy al|)"

consts 
    Alloc   :: "'a allocState_d program"
    Client  :: "'a clientState_d program"
    Network :: "'a systemState program"
    System  :: "'a systemState program"
  
axioms
    Alloc:   "Alloc   : alloc_spec"
    Client:  "Client  : client_spec"
    Network: "Network : network_spec"

defs
    System_def:
      "System == rename sysOfAlloc Alloc Join Network Join
                 (rename sysOfClient
                  (plam x: lessThan Nclients. rename client_map Client))"


(**
locale System =
  fixes 
    Alloc   :: 'a allocState_d program
    Client  :: 'a clientState_d program
    Network :: 'a systemState program
    System  :: 'a systemState program

  assumes
    Alloc   "Alloc   : alloc_spec"
    Client  "Client  : client_spec"
    Network "Network : network_spec"

  defines
    System_def
      "System == rename sysOfAlloc Alloc
                 Join
                 Network
                 Join
                 (rename sysOfClient
                  (plam x: lessThan Nclients. rename client_map Client))"
**)

ML {* use_legacy_bindings (the_context ()) *}

end

theorem inj_sysOfAlloc:

  inj sysOfAlloc

theorem inv_sysOfAlloc_eq:

  inv sysOfAlloc s =
  (| allocGiv = allocGiv s, allocAsk = allocAsk s, allocRel = allocRel s,
     allocState_d.dummy = (client s, systemState.dummy s) |)

theorem surj_sysOfAlloc:

  surj sysOfAlloc

theorem bij_sysOfAlloc:

  bij sysOfAlloc

theorem inj_sysOfClient:

  inj sysOfClient

theorem inv_sysOfClient_eq:

  inv sysOfClient s =
  (client s,
   (| allocGiv = allocGiv s, allocAsk = allocAsk s, allocRel = allocRel s,
      allocState_d.dummy = systemState.dummy s |))

theorem surj_sysOfClient:

  surj sysOfClient

theorem bij_sysOfClient:

  bij sysOfClient

theorem inj_client_map:

  inj client_map

theorem inv_client_map_eq:

  inv client_map s =
  (%(x, y). (| giv = giv x, ask = ask x, rel = rel x, clientState_d.dummy = y |))
   s

theorem surj_client_map:

  surj client_map

theorem bij_client_map:

  bij client_map

theorem fst_o_client_map:

  fst o client_map = non_dummy

theorem snd_o_client_map:

  snd o client_map = clientState_d.dummy

theorem client_o_sysOfAlloc:

  client o sysOfAlloc = fst o allocState_d.dummy

theorem allocGiv_o_sysOfAlloc_eq:

  allocGiv o sysOfAlloc = allocGiv

theorem allocAsk_o_sysOfAlloc_eq:

  allocAsk o sysOfAlloc = allocAsk

theorem allocRel_o_sysOfAlloc_eq:

  allocRel o sysOfAlloc = allocRel

theorem client_o_sysOfClient:

  client o sysOfClient = fst

theorem allocGiv_o_sysOfClient_eq:

  allocGiv o sysOfClient = allocGiv o snd

theorem allocAsk_o_sysOfClient_eq:

  allocAsk o sysOfClient = allocAsk o snd

theorem allocRel_o_sysOfClient_eq:

  allocRel o sysOfClient = allocRel o snd

theorem allocGiv_o_inv_sysOfAlloc_eq:

  allocGiv o inv sysOfAlloc = allocGiv

theorem allocAsk_o_inv_sysOfAlloc_eq:

  allocAsk o inv sysOfAlloc = allocAsk

theorem allocRel_o_inv_sysOfAlloc_eq:

  allocRel o inv sysOfAlloc = allocRel

theorem rel_inv_client_map_drop_map:

  rel o inv client_map o drop_map i o inv sysOfClient = rel o sub i o client

theorem ask_inv_client_map_drop_map:

  ask o inv client_map o drop_map i o inv sysOfClient = ask o sub i o client

theorem Network_component_System:

  Network Join
  (rename sysOfClient (plam x:{..<Nclients}. rename client_map Client) Join
   rename sysOfAlloc Alloc) =
  System

theorem Client_component_System:

  rename sysOfClient (plam x:{..<Nclients}. rename client_map Client) Join
  (Network Join rename sysOfAlloc Alloc) =
  System

theorem Alloc_component_System:

  rename sysOfAlloc Alloc Join
  (rename sysOfClient (plam x:{..<Nclients}. rename client_map Client) Join
   Network) =
  System

theorem Client_Allowed:

  Allowed Client = preserves rel ∩ preserves ask

theorem Network_Allowed:

  Allowed Network =
  preserves allocRel ∩ (INT i<Nclients. preserves (giv o sub i o client))

theorem Alloc_Allowed:

  Allowed Alloc = preserves allocGiv

theorem OK_lift_rename_Client:

  OK I (%i. lift i (rename client_map Client))