Theory Client

theory Client
imports AllocBase
(*  Title:      HOL/UNITY/Comp/Client.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1998  University of Cambridge

section‹Distributed Resource Management System: the Client›

theory Client imports "../Rename" AllocBase begin

  tokbag = nat     ― ‹tokbags could be multisets...or any ordered type?›

record state =
  giv :: "tokbag list" ― ‹input history: tokens granted›
  ask :: "tokbag list" ― ‹output history: tokens requested›
  rel :: "tokbag list" ― ‹output history: tokens released›
  tok :: tokbag        ― ‹current token request›

record 'a state_d =
  state +  
  dummy :: 'a          ― ‹new variables›

(*Array indexing is translated to list indexing as A[n] == A!(n-1). *)

  (** Release some tokens **)
  rel_act :: "('a state_d * 'a state_d) set"
  where "rel_act = {(s,s').
                  ∃nrel. nrel = size (rel s) &
                         s' = s (| rel := rel s @ [giv s!nrel] |) &
                         nrel < size (giv s) &
                         ask s!nrel ≤ giv s!nrel}"

  (** Choose a new token requirement **)

  (** Including s'=s suppresses fairness, allowing the non-trivial part
      of the action to be ignored **)

  tok_act :: "('a state_d * 'a state_d) set"
  where "tok_act = {(s,s'). s'=s | s' = s (|tok := Suc (tok s mod NbT) |)}"
  ask_act :: "('a state_d * 'a state_d) set"
  where "ask_act = {(s,s'). s'=s |
                         (s' = s (|ask := ask s @ [tok s]|))}"

  Client :: "'a state_d program"
  where "Client =
            ({s. tok s ∈ atMost NbT &
                 giv s = [] & ask s = [] & rel s = []},
             {rel_act, tok_act, ask_act},
             ⋃G ∈ preserves rel Int preserves ask Int preserves tok.
                   Acts G)"

  (*Maybe want a special theory section to declare such maps*)
  non_dummy :: "'a state_d => state"
  where "non_dummy s = (|giv = giv s, ask = ask s, rel = rel s, tok = tok s|)"

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

declare Client_def [THEN def_prg_Init, simp]
declare Client_def [THEN def_prg_AllowedActs, simp]
declare rel_act_def [THEN def_act_simp, simp]
declare tok_act_def [THEN def_act_simp, simp]
declare ask_act_def [THEN def_act_simp, simp]

lemma Client_ok_iff [iff]:
     "(Client ok G) =  
      (G ∈ preserves rel & G ∈ preserves ask & G ∈ preserves tok & 
       Client ∈ Allowed G)"
by (auto simp add: ok_iff_Allowed Client_def [THEN def_total_prg_Allowed])

text‹Safety property 1: ask, rel are increasing›
lemma increasing_ask_rel:
     "Client ∈ UNIV guarantees Increasing ask Int Increasing rel"
apply (auto intro!: increasing_imp_Increasing simp add: guar_def preserves_subset_increasing [THEN subsetD])
apply (auto simp add: Client_def increasing_def)
apply (safety, auto)+

declare nth_append [simp] append_one_prefix [simp]

text‹Safety property 2: the client never requests too many tokens.
      With no Substitution Axiom, we must prove the two invariants 
lemma ask_bounded_lemma:
     "Client ok G   
      ==> Client ⊔ G ∈   
              Always ({s. tok s ≤ NbT}  Int   
                      {s. ∀elt ∈ set (ask s). elt ≤ NbT})"
apply auto
apply (rule invariantI [THEN stable_Join_Always2], force) 
 prefer 2
 apply (fast elim!: preserves_subset_stable [THEN subsetD] intro!: stable_Int) 
apply (simp add: Client_def, safety)
apply (cut_tac m = "tok s" in NbT_pos [THEN mod_less_divisor], auto)

text‹export version, with no mention of tok in the postcondition, but
  unfortunately tok must be declared local.›
lemma ask_bounded:
     "Client ∈ UNIV guarantees Always {s. ∀elt ∈ set (ask s). elt ≤ NbT}"
apply (rule guaranteesI)
apply (erule ask_bounded_lemma [THEN Always_weaken])
apply (rule Int_lower2)

text‹** Towards proving the liveness property **›

lemma stable_rel_le_giv: "Client ∈ stable {s. rel s ≤ giv s}"
by (simp add: Client_def, safety, auto)

lemma Join_Stable_rel_le_giv:
     "[| Client ⊔ G ∈ Increasing giv;  G ∈ preserves rel |]  
      ==> Client ⊔ G ∈ Stable {s. rel s ≤ giv s}"
by (rule stable_rel_le_giv [THEN Increasing_preserves_Stable], auto)

lemma Join_Always_rel_le_giv:
     "[| Client ⊔ G ∈ Increasing giv;  G ∈ preserves rel |]  
      ==> Client ⊔ G ∈ Always {s. rel s ≤ giv s}"
by (force intro: AlwaysI Join_Stable_rel_le_giv)

lemma transient_lemma:
     "Client ∈ transient {s. rel s = k & k<h & h ≤ giv s & h pfixGe ask s}"
apply (simp add: Client_def mk_total_program_def)
apply (rule_tac act = rel_act in totalize_transientI)
  apply (auto simp add: Domain_unfold Client_def)
 apply (blast intro: less_le_trans prefix_length_le strict_prefix_length_less)
apply (auto simp add: prefix_def genPrefix_iff_nth Ge_def)
apply (blast intro: strict_prefix_length_less)

lemma induct_lemma:
     "[| Client ⊔ G ∈ Increasing giv;  Client ok G |]  
  ==> Client ⊔ G ∈ {s. rel s = k & k<h & h ≤ giv s & h pfixGe ask s}   
                      LeadsTo {s. k < rel s & rel s ≤ giv s &  
                                  h ≤ giv s & h pfixGe ask s}"
apply (rule single_LeadsTo_I)
apply (frule increasing_ask_rel [THEN guaranteesD], auto)
apply (rule transient_lemma [THEN Join_transient_I1, THEN transient_imp_leadsTo, THEN leadsTo_imp_LeadsTo, THEN PSP_Stable, THEN LeadsTo_weaken])
  apply (rule Stable_Int [THEN Stable_Int, THEN Stable_Int])
     apply (erule_tac f = giv and x = "giv s" in IncreasingD)
    apply (erule_tac f = ask and x = "ask s" in IncreasingD)
   apply (erule_tac f = rel and x = "rel s" in IncreasingD)
  apply (erule Join_Stable_rel_le_giv, blast)
 apply (blast intro: order_less_imp_le order_trans)
apply (blast intro: sym order_less_le [THEN iffD2] order_trans
                    prefix_imp_pfixGe pfixGe_trans)

lemma rel_progress_lemma:
     "[| Client ⊔ G ∈ Increasing giv;  Client ok G |]  
  ==> Client ⊔ G ∈ {s. rel s < h & h ≤ giv s & h pfixGe ask s}   
                      LeadsTo {s. h ≤ rel s}"
apply (rule_tac f = "%s. size h - size (rel s) " in LessThan_induct)
apply (auto simp add: vimage_def)
apply (rule single_LeadsTo_I)
apply (rule induct_lemma [THEN LeadsTo_weaken], auto)
 apply (blast intro: order_less_le [THEN iffD2] dest: common_prefix_linear)
apply (drule strict_prefix_length_less)+
apply arith

lemma client_progress_lemma:
     "[| Client ⊔ G ∈ Increasing giv;  Client ok G |]  
      ==> Client ⊔ G ∈ {s. h ≤ giv s & h pfixGe ask s}   
                          LeadsTo {s. h ≤ rel s}"
apply (rule Join_Always_rel_le_giv [THEN Always_LeadsToI], simp_all) 
apply (rule LeadsTo_Un [THEN LeadsTo_weaken_L])
  apply (blast intro: rel_progress_lemma) 
 apply (rule subset_refl [THEN subset_imp_LeadsTo])
apply (blast intro: order_less_le [THEN iffD2] dest: common_prefix_linear)

text‹Progress property: all tokens that are given will be released›
lemma client_progress:
     "Client ∈  
       Increasing giv  guarantees   
       (INT h. {s. h ≤ giv s & h pfixGe ask s} LeadsTo {s. h ≤ rel s})"
apply (rule guaranteesI, clarify)
apply (blast intro: client_progress_lemma)

text‹This shows that the Client won't alter other variables in any state
  that it is combined with›
lemma client_preserves_dummy: "Client ∈ preserves dummy"
by (simp add: Client_def preserves_def, clarify, safety, auto)

text‹* Obsolete lemmas from first version of the Client *›

lemma stable_size_rel_le_giv:
     "Client ∈ stable {s. size (rel s) ≤ size (giv s)}"
by (simp add: Client_def, safety, auto)

text‹clients return the right number of tokens›
lemma ok_guar_rel_prefix_giv:
     "Client ∈ Increasing giv  guarantees  Always {s. rel s ≤ giv s}"
apply (rule guaranteesI)
apply (rule AlwaysI, force)
apply (blast intro: Increasing_preserves_Stable stable_rel_le_giv)