author | wenzelm |
Sat, 02 Dec 2006 02:52:02 +0100 | |
changeset 21624 | 6f79647cf536 |
parent 17309 | c43ed29bd197 |
child 24180 | 9f818139951b |
permissions | -rw-r--r-- |
3807 | 1 |
(* |
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File: MemoryImplementation.thy |
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ID: $Id$ |
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Author: Stephan Merz |
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Copyright: 1997 University of Munich |
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*) |
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header {* RPC-Memory example: Memory implementation *} |
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theory MemoryImplementation |
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imports Memory RPC MemClerk |
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begin |
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datatype histState = histA | histB |
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types |
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histType = "(PrIds => histState) stfun" (* the type of the history variable *) |
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consts |
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(* the specification *) |
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(* channel (external) *) |
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memCh :: "memChType" |
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(* internal variables *) |
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mm :: "memType" |
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(* the state variables of the implementation *) |
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(* channels *) |
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(* same interface channel memCh *) |
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crCh :: "rpcSndChType" |
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rmCh :: "rpcRcvChType" |
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(* internal variables *) |
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(* identity refinement mapping for mm -- simply reused *) |
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rst :: "rpcStType" |
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cst :: "mClkStType" |
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ires :: "resType" |
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||
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constdefs |
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(* auxiliary predicates *) |
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MVOKBARF :: "Vals => bool" |
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"MVOKBARF v == (v : MemVal) | (v = OK) | (v = BadArg) | (v = RPCFailure)" |
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MVOKBA :: "Vals => bool" |
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"MVOKBA v == (v : MemVal) | (v = OK) | (v = BadArg)" |
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MVNROKBA :: "Vals => bool" |
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"MVNROKBA v == (v : MemVal) | (v = NotAResult) | (v = OK) | (v = BadArg)" |
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(* tuples of state functions changed by the various components *) |
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e :: "PrIds => (bit * memOp) stfun" |
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"e p == PRED (caller memCh!p)" |
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c :: "PrIds => (mClkState * (bit * Vals) * (bit * rpcOp)) stfun" |
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"c p == PRED (cst!p, rtrner memCh!p, caller crCh!p)" |
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r :: "PrIds => (rpcState * (bit * Vals) * (bit * memOp)) stfun" |
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"r p == PRED (rst!p, rtrner crCh!p, caller rmCh!p)" |
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m :: "PrIds => ((bit * Vals) * Vals) stfun" |
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"m p == PRED (rtrner rmCh!p, ires!p)" |
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||
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(* the environment action *) |
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ENext :: "PrIds => action" |
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"ENext p == ACT (? l. #l : #MemLoc & Call memCh p #(read l))" |
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(* specification of the history variable *) |
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HInit :: "histType => PrIds => stpred" |
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"HInit rmhist p == PRED rmhist!p = #histA" |
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||
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HNext :: "histType => PrIds => action" |
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"HNext rmhist p == ACT (rmhist!p)$ = |
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(if (MemReturn rmCh ires p | RPCFail crCh rmCh rst p) |
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then #histB |
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else if (MClkReply memCh crCh cst p) |
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then #histA |
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else $(rmhist!p))" |
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||
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HistP :: "histType => PrIds => temporal" |
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"HistP rmhist p == TEMP Init HInit rmhist p |
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& [][HNext rmhist p]_(c p,r p,m p, rmhist!p)" |
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Hist :: "histType => temporal" |
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"Hist rmhist == TEMP (ALL p. HistP rmhist p)" |
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(* the implementation *) |
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IPImp :: "PrIds => temporal" |
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"IPImp p == TEMP ( Init ~Calling memCh p & [][ENext p]_(e p) |
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& MClkIPSpec memCh crCh cst p |
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& RPCIPSpec crCh rmCh rst p |
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& RPSpec rmCh mm ires p |
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& (ALL l. #l : #MemLoc --> MSpec rmCh mm ires l))" |
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ImpInit :: "PrIds => stpred" |
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"ImpInit p == PRED ( ~Calling memCh p |
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& MClkInit crCh cst p |
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& RPCInit rmCh rst p |
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& PInit ires p)" |
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ImpNext :: "PrIds => action" |
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"ImpNext p == ACT [ENext p]_(e p) |
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& [MClkNext memCh crCh cst p]_(c p) |
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& [RPCNext crCh rmCh rst p]_(r p) |
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& [RNext rmCh mm ires p]_(m p)" |
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ImpLive :: "PrIds => temporal" |
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"ImpLive p == TEMP WF(MClkFwd memCh crCh cst p)_(c p) |
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& SF(MClkReply memCh crCh cst p)_(c p) |
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& WF(RPCNext crCh rmCh rst p)_(r p) |
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& WF(RNext rmCh mm ires p)_(m p) |
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& WF(MemReturn rmCh ires p)_(m p)" |
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Implementation :: "temporal" |
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"Implementation == TEMP ( (ALL p. Init (~Calling memCh p) & [][ENext p]_(e p)) |
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& MClkISpec memCh crCh cst |
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& RPCISpec crCh rmCh rst |
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& IRSpec rmCh mm ires)" |
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(* the predicate S describes the states of the implementation. |
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slight simplification: two "histState" parameters instead of a |
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(one- or two-element) set. |
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NB: The second conjunct of the definition in the paper is taken care of by |
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the type definitions. The last conjunct is asserted separately as the memory |
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invariant MemInv, proved in Memory.ML. *) |
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S :: "histType => bool => bool => bool => mClkState => rpcState => histState => histState => PrIds => stpred" |
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"S rmhist ecalling ccalling rcalling cs rs hs1 hs2 p == PRED |
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Calling memCh p = #ecalling |
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& Calling crCh p = #ccalling |
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& (#ccalling --> arg<crCh!p> = MClkRelayArg<arg<memCh!p>>) |
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& (~ #ccalling & cst!p = #clkB --> MVOKBARF<res<crCh!p>>) |
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& Calling rmCh p = #rcalling |
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& (#rcalling --> arg<rmCh!p> = RPCRelayArg<arg<crCh!p>>) |
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& (~ #rcalling --> ires!p = #NotAResult) |
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& (~ #rcalling & rst!p = #rpcB --> MVOKBA<res<rmCh!p>>) |
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& cst!p = #cs |
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& rst!p = #rs |
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& (rmhist!p = #hs1 | rmhist!p = #hs2) |
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& MVNROKBA<ires!p>" |
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(* predicates S1 -- S6 define special instances of S *) |
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S1 :: "histType => PrIds => stpred" |
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"S1 rmhist p == S rmhist False False False clkA rpcA histA histA p" |
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S2 :: "histType => PrIds => stpred" |
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"S2 rmhist p == S rmhist True False False clkA rpcA histA histA p" |
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S3 :: "histType => PrIds => stpred" |
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"S3 rmhist p == S rmhist True True False clkB rpcA histA histB p" |
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S4 :: "histType => PrIds => stpred" |
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"S4 rmhist p == S rmhist True True True clkB rpcB histA histB p" |
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S5 :: "histType => PrIds => stpred" |
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"S5 rmhist p == S rmhist True True False clkB rpcB histB histB p" |
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S6 :: "histType => PrIds => stpred" |
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"S6 rmhist p == S rmhist True False False clkB rpcA histB histB p" |
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(* The invariant asserts that the system is always in one of S1 - S6, for every p *) |
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ImpInv :: "histType => PrIds => stpred" |
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"ImpInv rmhist p == PRED ( S1 rmhist p | S2 rmhist p | S3 rmhist p |
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| S4 rmhist p | S5 rmhist p | S6 rmhist p)" |
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resbar :: "histType => resType" (* refinement mapping *) |
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"resbar rmhist s p == |
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(if (S1 rmhist p s | S2 rmhist p s) |
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then ires s p |
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else if S3 rmhist p s |
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then if rmhist s p = histA |
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then ires s p else MemFailure |
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else if S4 rmhist p s |
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then if (rmhist s p = histB & ires s p = NotAResult) |
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then MemFailure else ires s p |
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else if S5 rmhist p s |
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then res (rmCh s p) |
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else if S6 rmhist p s |
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then if res (crCh s p) = RPCFailure |
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then MemFailure else res (crCh s p) |
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else NotAResult)" (* dummy value *) |
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axioms |
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(* the "base" variables: everything except resbar and hist (for any index) *) |
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MI_base: "basevars (caller memCh!p, |
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(rtrner memCh!p, caller crCh!p, cst!p), |
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(rtrner crCh!p, caller rmCh!p, rst!p), |
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(mm!l, rtrner rmCh!p, ires!p))" |
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(* |
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The main theorem is theorem "Implementation" at the end of this file, |
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which shows that the composition of a reliable memory, an RPC component, and |
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a memory clerk implements an unreliable memory. The files "MIsafe.ML" and |
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"MIlive.ML" contain lower-level lemmas for the safety and liveness parts. |
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Steps are (roughly) numbered as in the hand proof. |
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*) |
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(* --------------------------- automatic prover --------------------------- *) |
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declare if_weak_cong [cong del] |
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ML {* val MI_css = (claset(), simpset()) *} |
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(* A more aggressive variant that tries to solve subgoals by assumption |
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or contradiction during the simplification. |
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THIS IS UNSAFE, BECAUSE IT DOESN'T RECORD THE CHOICES!! |
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(but it can be a lot faster than MI_css) |
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*) |
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ML {* |
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val MI_fast_css = |
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let |
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val (cs,ss) = MI_css |
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in |
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(cs addSEs [temp_use (thm "squareE")], |
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ss addSSolver (mk_solver "" (fn thms => assume_tac ORELSE' (etac notE)))) |
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end; |
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val temp_elim = make_elim o temp_use; |
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*} |
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(****************************** The history variable ******************************) |
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section "History variable" |
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lemma HistoryLemma: "|- Init(ALL p. ImpInit p) & [](ALL p. ImpNext p) |
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--> (EEX rmhist. Init(ALL p. HInit rmhist p) |
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& [](ALL p. [HNext rmhist p]_(c p, r p, m p, rmhist!p)))" |
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apply clarsimp |
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apply (rule historyI) |
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apply assumption+ |
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apply (rule MI_base) |
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apply (tactic {* action_simp_tac (simpset () addsimps [thm "HInit_def"]) [] [] 1 *}) |
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apply (erule fun_cong) |
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apply (tactic {* action_simp_tac (simpset () addsimps [thm "HNext_def"]) |
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[thm "busy_squareI"] [] 1 *}) |
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apply (erule fun_cong) |
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done |
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lemma History: "|- Implementation --> (EEX rmhist. Hist rmhist)" |
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apply clarsimp |
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apply (rule HistoryLemma [temp_use, THEN eex_mono]) |
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prefer 3 |
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apply (force simp: Hist_def HistP_def Init_def all_box [try_rewrite] |
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split_box_conj [try_rewrite]) |
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apply (auto simp: Implementation_def MClkISpec_def RPCISpec_def |
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IRSpec_def MClkIPSpec_def RPCIPSpec_def RPSpec_def ImpInit_def |
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Init_def ImpNext_def c_def r_def m_def all_box [temp_use] split_box_conj [temp_use]) |
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done |
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(******************************** The safety part *********************************) |
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section "The safety part" |
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(* ------------------------- Include lower-level lemmas ------------------------- *) |
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(* RPCFailure notin MemVals U {OK,BadArg} *) |
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lemma MVOKBAnotRF: "MVOKBA x ==> x ~= RPCFailure" |
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apply (unfold MVOKBA_def) |
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apply auto |
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done |
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(* NotAResult notin MemVals U {OK,BadArg,RPCFailure} *) |
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lemma MVOKBARFnotNR: "MVOKBARF x ==> x ~= NotAResult" |
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apply (unfold MVOKBARF_def) |
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apply auto |
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done |
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(* ================ Si's are mutually exclusive ================================ *) |
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(* Si and Sj are mutually exclusive for i # j. This helps to simplify the big |
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conditional in the definition of resbar when doing the step-simulation proof. |
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We prove a weaker result, which suffices for our purposes: |
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Si implies (not Sj), for j<i. |
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*) |
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(* --- not used --- |
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Goal "|- S1 rmhist p --> S1 rmhist p & ~S2 rmhist p & ~S3 rmhist p & |
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~S4 rmhist p & ~S5 rmhist p & ~S6 rmhist p" |
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by (auto_tac (MI_css addsimps2 [S_def, S1_def, S2_def, |
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S3_def, S4_def, S5_def, S6_def])); |
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qed "S1_excl"; |
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*) |
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lemma S2_excl: "|- S2 rmhist p --> S2 rmhist p & ~S1 rmhist p" |
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by (auto simp: S_def S1_def S2_def) |
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lemma S3_excl: "|- S3 rmhist p --> S3 rmhist p & ~S1 rmhist p & ~S2 rmhist p" |
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by (auto simp: S_def S1_def S2_def S3_def) |
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lemma S4_excl: "|- S4 rmhist p --> S4 rmhist p & ~S1 rmhist p & ~S2 rmhist p & ~S3 rmhist p" |
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by (auto simp: S_def S1_def S2_def S3_def S4_def) |
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lemma S5_excl: "|- S5 rmhist p --> S5 rmhist p & ~S1 rmhist p & ~S2 rmhist p |
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& ~S3 rmhist p & ~S4 rmhist p" |
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by (auto simp: S_def S1_def S2_def S3_def S4_def S5_def) |
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lemma S6_excl: "|- S6 rmhist p --> S6 rmhist p & ~S1 rmhist p & ~S2 rmhist p |
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& ~S3 rmhist p & ~S4 rmhist p & ~S5 rmhist p" |
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by (auto simp: S_def S1_def S2_def S3_def S4_def S5_def S6_def) |
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(* ==================== Lemmas about the environment ============================== *) |
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lemma Envbusy: "|- $(Calling memCh p) --> ~ENext p" |
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by (auto simp: ENext_def Call_def) |
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(* ==================== Lemmas about the implementation's states ==================== *) |
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300 |
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(* The following series of lemmas are used in establishing the implementation's |
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next-state relation (Step 1.2 of the proof in the paper). For each state Si, we |
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determine which component actions are possible and what state they result in. |
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*) |
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(* ------------------------------ State S1 ---------------------------------------- *) |
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307 |
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lemma S1Env: "|- ENext p & $(S1 rmhist p) & unchanged (c p, r p, m p, rmhist!p) |
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--> (S2 rmhist p)$" |
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by (force simp: ENext_def Call_def c_def r_def m_def |
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caller_def rtrner_def MVNROKBA_def S_def S1_def S2_def Calling_def) |
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lemma S1ClerkUnch: "|- [MClkNext memCh crCh cst p]_(c p) & $(S1 rmhist p) --> unchanged (c p)" |
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by (tactic {* auto_tac (MI_fast_css addSDs2 [temp_use (thm "MClkidle")] |
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addsimps2 [thm "S_def", thm "S1_def"]) *}) |
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lemma S1RPCUnch: "|- [RPCNext crCh rmCh rst p]_(r p) & $(S1 rmhist p) --> unchanged (r p)" |
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by (tactic {* auto_tac (MI_fast_css addSDs2 [temp_use (thm "RPCidle")] |
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addsimps2 [thm "S_def", thm "S1_def"]) *}) |
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320 |
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lemma S1MemUnch: "|- [RNext rmCh mm ires p]_(m p) & $(S1 rmhist p) --> unchanged (m p)" |
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322 |
by (tactic {* auto_tac (MI_fast_css addSDs2 [temp_use (thm "Memoryidle")] |
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addsimps2 [thm "S_def", thm "S1_def"]) *}) |
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324 |
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lemma S1Hist: "|- [HNext rmhist p]_(c p,r p,m p,rmhist!p) & $(S1 rmhist p) |
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--> unchanged (rmhist!p)" |
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327 |
by (tactic {* action_simp_tac (simpset () addsimps [thm "HNext_def", thm "S_def", |
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328 |
thm "S1_def", thm "MemReturn_def", thm "RPCFail_def", thm "MClkReply_def", |
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thm "Return_def"]) [] [temp_use (thm "squareE")] 1 *}) |
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330 |
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331 |
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332 |
(* ------------------------------ State S2 ---------------------------------------- *) |
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333 |
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334 |
lemma S2EnvUnch: "|- [ENext p]_(e p) & $(S2 rmhist p) --> unchanged (e p)" |
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335 |
by (auto dest!: Envbusy [temp_use] simp: S_def S2_def) |
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336 |
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337 |
lemma S2Clerk: "|- MClkNext memCh crCh cst p & $(S2 rmhist p) --> MClkFwd memCh crCh cst p" |
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338 |
by (auto simp: MClkNext_def MClkRetry_def MClkReply_def S_def S2_def) |
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339 |
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340 |
lemma S2Forward: "|- $(S2 rmhist p) & MClkFwd memCh crCh cst p |
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341 |
& unchanged (e p, r p, m p, rmhist!p) |
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342 |
--> (S3 rmhist p)$" |
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343 |
by (tactic {* action_simp_tac (simpset () addsimps [thm "MClkFwd_def", |
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344 |
thm "Call_def", thm "e_def", thm "r_def", thm "m_def", thm "caller_def", |
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thm "rtrner_def", thm "S_def", thm "S2_def", thm "S3_def", thm "Calling_def"]) [] [] 1 *}) |
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347 |
lemma S2RPCUnch: "|- [RPCNext crCh rmCh rst p]_(r p) & $(S2 rmhist p) --> unchanged (r p)" |
|
348 |
by (auto simp: S_def S2_def dest!: RPCidle [temp_use]) |
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349 |
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350 |
lemma S2MemUnch: "|- [RNext rmCh mm ires p]_(m p) & $(S2 rmhist p) --> unchanged (m p)" |
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351 |
by (auto simp: S_def S2_def dest!: Memoryidle [temp_use]) |
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353 |
lemma S2Hist: "|- [HNext rmhist p]_(c p,r p,m p,rmhist!p) & $(S2 rmhist p) |
|
354 |
--> unchanged (rmhist!p)" |
|
355 |
by (tactic {* auto_tac (MI_fast_css addsimps2 [thm "HNext_def", thm "MemReturn_def", |
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356 |
thm "RPCFail_def", thm "MClkReply_def", thm "Return_def", thm "S_def", thm "S2_def"]) *}) |
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357 |
||
358 |
(* ------------------------------ State S3 ---------------------------------------- *) |
|
359 |
||
360 |
lemma S3EnvUnch: "|- [ENext p]_(e p) & $(S3 rmhist p) --> unchanged (e p)" |
|
361 |
by (auto dest!: Envbusy [temp_use] simp: S_def S3_def) |
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362 |
||
363 |
lemma S3ClerkUnch: "|- [MClkNext memCh crCh cst p]_(c p) & $(S3 rmhist p) --> unchanged (c p)" |
|
364 |
by (auto dest!: MClkbusy [temp_use] simp: square_def S_def S3_def) |
|
365 |
||
366 |
lemma S3LegalRcvArg: "|- S3 rmhist p --> IsLegalRcvArg<arg<crCh!p>>" |
|
367 |
by (auto simp: IsLegalRcvArg_def MClkRelayArg_def S_def S3_def) |
|
368 |
||
369 |
lemma S3RPC: "|- RPCNext crCh rmCh rst p & $(S3 rmhist p) |
|
370 |
--> RPCFwd crCh rmCh rst p | RPCFail crCh rmCh rst p" |
|
371 |
apply clarsimp |
|
372 |
apply (frule S3LegalRcvArg [action_use]) |
|
373 |
apply (auto simp: RPCNext_def RPCReject_def RPCReply_def S_def S3_def) |
|
374 |
done |
|
375 |
||
376 |
lemma S3Forward: "|- RPCFwd crCh rmCh rst p & HNext rmhist p & $(S3 rmhist p) |
|
377 |
& unchanged (e p, c p, m p) |
|
378 |
--> (S4 rmhist p)$ & unchanged (rmhist!p)" |
|
379 |
by (tactic {* action_simp_tac (simpset () addsimps [thm "RPCFwd_def", |
|
380 |
thm "HNext_def", thm "MemReturn_def", thm "RPCFail_def", |
|
381 |
thm "MClkReply_def", thm "Return_def", thm "Call_def", thm "e_def", |
|
382 |
thm "c_def", thm "m_def", thm "caller_def", thm "rtrner_def", thm "S_def", |
|
383 |
thm "S3_def", thm "S4_def", thm "Calling_def"]) [] [] 1 *}) |
|
384 |
||
385 |
lemma S3Fail: "|- RPCFail crCh rmCh rst p & $(S3 rmhist p) & HNext rmhist p |
|
386 |
& unchanged (e p, c p, m p) |
|
387 |
--> (S6 rmhist p)$" |
|
388 |
by (tactic {* action_simp_tac (simpset () addsimps [thm "HNext_def", |
|
389 |
thm "RPCFail_def", thm "Return_def", thm "e_def", thm "c_def", |
|
390 |
thm "m_def", thm "caller_def", thm "rtrner_def", thm "MVOKBARF_def", |
|
391 |
thm "S_def", thm "S3_def", thm "S6_def", thm "Calling_def"]) [] [] 1 *}) |
|
392 |
||
393 |
lemma S3MemUnch: "|- [RNext rmCh mm ires p]_(m p) & $(S3 rmhist p) --> unchanged (m p)" |
|
394 |
by (auto simp: S_def S3_def dest!: Memoryidle [temp_use]) |
|
395 |
||
396 |
lemma S3Hist: "|- HNext rmhist p & $(S3 rmhist p) & unchanged (r p) --> unchanged (rmhist!p)" |
|
397 |
by (auto simp: HNext_def MemReturn_def RPCFail_def MClkReply_def |
|
398 |
Return_def r_def rtrner_def S_def S3_def Calling_def) |
|
399 |
||
400 |
(* ------------------------------ State S4 ---------------------------------------- *) |
|
401 |
||
402 |
lemma S4EnvUnch: "|- [ENext p]_(e p) & $(S4 rmhist p) --> unchanged (e p)" |
|
403 |
by (auto simp: S_def S4_def dest!: Envbusy [temp_use]) |
|
404 |
||
405 |
lemma S4ClerkUnch: "|- [MClkNext memCh crCh cst p]_(c p) & $(S4 rmhist p) --> unchanged (c p)" |
|
406 |
by (auto simp: S_def S4_def dest!: MClkbusy [temp_use]) |
|
407 |
||
408 |
lemma S4RPCUnch: "|- [RPCNext crCh rmCh rst p]_(r p) & $(S4 rmhist p) --> unchanged (r p)" |
|
409 |
by (tactic {* auto_tac (MI_fast_css addsimps2 [thm "S_def", thm "S4_def"] |
|
410 |
addSDs2 [temp_use (thm "RPCbusy")]) *}) |
|
411 |
||
412 |
lemma S4ReadInner: "|- ReadInner rmCh mm ires p l & $(S4 rmhist p) & unchanged (e p, c p, r p) |
|
413 |
& HNext rmhist p & $(MemInv mm l) |
|
414 |
--> (S4 rmhist p)$ & unchanged (rmhist!p)" |
|
415 |
by (tactic {* action_simp_tac (simpset () addsimps [thm "ReadInner_def", |
|
416 |
thm "GoodRead_def", thm "BadRead_def", thm "HNext_def", thm "MemReturn_def", |
|
417 |
thm "RPCFail_def", thm "MClkReply_def", thm "Return_def", thm "e_def", |
|
418 |
thm "c_def", thm "r_def", thm "rtrner_def", thm "caller_def", |
|
419 |
thm "MVNROKBA_def", thm "S_def", thm "S4_def", thm "RdRequest_def", |
|
420 |
thm "Calling_def", thm "MemInv_def"]) [] [] 1 *}) |
|
421 |
||
422 |
lemma S4Read: "|- Read rmCh mm ires p & $(S4 rmhist p) & unchanged (e p, c p, r p) |
|
423 |
& HNext rmhist p & (!l. $MemInv mm l) |
|
424 |
--> (S4 rmhist p)$ & unchanged (rmhist!p)" |
|
425 |
by (auto simp: Read_def dest!: S4ReadInner [temp_use]) |
|
426 |
||
427 |
lemma S4WriteInner: "|- WriteInner rmCh mm ires p l v & $(S4 rmhist p) & unchanged (e p, c p, r p) & HNext rmhist p |
|
428 |
--> (S4 rmhist p)$ & unchanged (rmhist!p)" |
|
429 |
by (tactic {* action_simp_tac (simpset () addsimps [thm "WriteInner_def", |
|
430 |
thm "GoodWrite_def", thm "BadWrite_def", thm "HNext_def", thm "MemReturn_def", |
|
431 |
thm "RPCFail_def", thm "MClkReply_def", thm "Return_def", thm "e_def", |
|
432 |
thm "c_def", thm "r_def", thm "rtrner_def", thm "caller_def", thm "MVNROKBA_def", |
|
433 |
thm "S_def", thm "S4_def", thm "WrRequest_def", thm "Calling_def"]) [] [] 1 *}) |
|
434 |
||
435 |
lemma S4Write: "|- Write rmCh mm ires p l & $(S4 rmhist p) & unchanged (e p, c p, r p) |
|
436 |
& (HNext rmhist p) |
|
437 |
--> (S4 rmhist p)$ & unchanged (rmhist!p)" |
|
438 |
by (auto simp: Write_def dest!: S4WriteInner [temp_use]) |
|
439 |
||
440 |
lemma WriteS4: "|- $ImpInv rmhist p & Write rmCh mm ires p l --> $S4 rmhist p" |
|
441 |
by (auto simp: Write_def WriteInner_def ImpInv_def |
|
442 |
WrRequest_def S_def S1_def S2_def S3_def S4_def S5_def S6_def) |
|
443 |
||
444 |
lemma S4Return: "|- MemReturn rmCh ires p & $S4 rmhist p & unchanged (e p, c p, r p) |
|
445 |
& HNext rmhist p |
|
446 |
--> (S5 rmhist p)$" |
|
447 |
by (auto simp: HNext_def MemReturn_def Return_def e_def c_def r_def |
|
448 |
rtrner_def caller_def MVNROKBA_def MVOKBA_def S_def S4_def S5_def Calling_def) |
|
449 |
||
450 |
lemma S4Hist: "|- HNext rmhist p & $S4 rmhist p & (m p)$ = $(m p) --> (rmhist!p)$ = $(rmhist!p)" |
|
451 |
by (auto simp: HNext_def MemReturn_def RPCFail_def MClkReply_def |
|
452 |
Return_def m_def rtrner_def S_def S4_def Calling_def) |
|
453 |
||
454 |
(* ------------------------------ State S5 ---------------------------------------- *) |
|
455 |
||
456 |
lemma S5EnvUnch: "|- [ENext p]_(e p) & $(S5 rmhist p) --> unchanged (e p)" |
|
457 |
by (auto simp: S_def S5_def dest!: Envbusy [temp_use]) |
|
458 |
||
459 |
lemma S5ClerkUnch: "|- [MClkNext memCh crCh cst p]_(c p) & $(S5 rmhist p) --> unchanged (c p)" |
|
460 |
by (auto simp: S_def S5_def dest!: MClkbusy [temp_use]) |
|
461 |
||
462 |
lemma S5RPC: "|- RPCNext crCh rmCh rst p & $(S5 rmhist p) |
|
463 |
--> RPCReply crCh rmCh rst p | RPCFail crCh rmCh rst p" |
|
464 |
by (auto simp: RPCNext_def RPCReject_def RPCFwd_def S_def S5_def) |
|
465 |
||
466 |
lemma S5Reply: "|- RPCReply crCh rmCh rst p & $(S5 rmhist p) & unchanged (e p, c p, m p,rmhist!p) |
|
467 |
--> (S6 rmhist p)$" |
|
468 |
by (tactic {* action_simp_tac (simpset () addsimps [thm "RPCReply_def", |
|
469 |
thm "Return_def", thm "e_def", thm "c_def", thm "m_def", thm "MVOKBA_def", |
|
470 |
thm "MVOKBARF_def", thm "caller_def", thm "rtrner_def", thm "S_def", |
|
471 |
thm "S5_def", thm "S6_def", thm "Calling_def"]) [] [] 1 *}) |
|
472 |
||
473 |
lemma S5Fail: "|- RPCFail crCh rmCh rst p & $(S5 rmhist p) & unchanged (e p, c p, m p,rmhist!p) |
|
474 |
--> (S6 rmhist p)$" |
|
475 |
by (tactic {* action_simp_tac (simpset () addsimps [thm "RPCFail_def", |
|
476 |
thm "Return_def", thm "e_def", thm "c_def", thm "m_def", |
|
477 |
thm "MVOKBARF_def", thm "caller_def", thm "rtrner_def", |
|
478 |
thm "S_def", thm "S5_def", thm "S6_def", thm "Calling_def"]) [] [] 1 *}) |
|
479 |
||
480 |
lemma S5MemUnch: "|- [RNext rmCh mm ires p]_(m p) & $(S5 rmhist p) --> unchanged (m p)" |
|
481 |
by (auto simp: S_def S5_def dest!: Memoryidle [temp_use]) |
|
482 |
||
483 |
lemma S5Hist: "|- [HNext rmhist p]_(c p, r p, m p, rmhist!p) & $(S5 rmhist p) |
|
484 |
--> (rmhist!p)$ = $(rmhist!p)" |
|
485 |
by (tactic {* auto_tac (MI_fast_css addsimps2 [thm "HNext_def", |
|
486 |
thm "MemReturn_def", thm "RPCFail_def", thm "MClkReply_def", thm "Return_def", |
|
487 |
thm "S_def", thm "S5_def"]) *}) |
|
488 |
||
489 |
(* ------------------------------ State S6 ---------------------------------------- *) |
|
490 |
||
491 |
lemma S6EnvUnch: "|- [ENext p]_(e p) & $(S6 rmhist p) --> unchanged (e p)" |
|
492 |
by (auto simp: S_def S6_def dest!: Envbusy [temp_use]) |
|
493 |
||
494 |
lemma S6Clerk: "|- MClkNext memCh crCh cst p & $(S6 rmhist p) |
|
495 |
--> MClkRetry memCh crCh cst p | MClkReply memCh crCh cst p" |
|
496 |
by (auto simp: MClkNext_def MClkFwd_def S_def S6_def) |
|
497 |
||
498 |
lemma S6Retry: "|- MClkRetry memCh crCh cst p & HNext rmhist p & $S6 rmhist p |
|
499 |
& unchanged (e p,r p,m p) |
|
500 |
--> (S3 rmhist p)$ & unchanged (rmhist!p)" |
|
501 |
by (tactic {* action_simp_tac (simpset () addsimps [thm "HNext_def", |
|
502 |
thm "MClkReply_def", thm "MClkRetry_def", thm "Call_def", thm "Return_def", |
|
503 |
thm "e_def", thm "r_def", thm "m_def", thm "caller_def", thm "rtrner_def", |
|
504 |
thm "S_def", thm "S6_def", thm "S3_def", thm "Calling_def"]) [] [] 1 *}) |
|
505 |
||
506 |
lemma S6Reply: "|- MClkReply memCh crCh cst p & HNext rmhist p & $S6 rmhist p |
|
507 |
& unchanged (e p,r p,m p) |
|
508 |
--> (S1 rmhist p)$" |
|
509 |
by (tactic {* action_simp_tac (simpset () addsimps [thm "HNext_def", |
|
510 |
thm "MemReturn_def", thm "RPCFail_def", thm "Return_def", thm "MClkReply_def", |
|
511 |
thm "e_def", thm "r_def", thm "m_def", thm "caller_def", thm "rtrner_def", |
|
512 |
thm "S_def", thm "S6_def", thm "S1_def", thm "Calling_def"]) [] [] 1 *}) |
|
513 |
||
514 |
lemma S6RPCUnch: "|- [RPCNext crCh rmCh rst p]_(r p) & $S6 rmhist p --> unchanged (r p)" |
|
515 |
by (auto simp: S_def S6_def dest!: RPCidle [temp_use]) |
|
516 |
||
517 |
lemma S6MemUnch: "|- [RNext rmCh mm ires p]_(m p) & $(S6 rmhist p) --> unchanged (m p)" |
|
518 |
by (auto simp: S_def S6_def dest!: Memoryidle [temp_use]) |
|
519 |
||
520 |
lemma S6Hist: "|- HNext rmhist p & $S6 rmhist p & (c p)$ = $(c p) --> (rmhist!p)$ = $(rmhist!p)" |
|
521 |
by (auto simp: HNext_def MClkReply_def Return_def c_def rtrner_def S_def S6_def Calling_def) |
|
522 |
||
523 |
||
524 |
section "Correctness of predicate-action diagram" |
|
525 |
||
526 |
||
527 |
(* ========== Step 1.1 ================================================= *) |
|
528 |
(* The implementation's initial condition implies the state predicate S1 *) |
|
529 |
||
530 |
lemma Step1_1: "|- ImpInit p & HInit rmhist p --> S1 rmhist p" |
|
531 |
by (tactic {* auto_tac (MI_fast_css addsimps2 [thm "MVNROKBA_def", |
|
532 |
thm "MClkInit_def", thm "RPCInit_def", thm "PInit_def", thm "HInit_def", |
|
533 |
thm "ImpInit_def", thm "S_def", thm "S1_def"]) *}) |
|
534 |
||
535 |
(* ========== Step 1.2 ================================================== *) |
|
536 |
(* Figure 16 is a predicate-action diagram for the implementation. *) |
|
537 |
||
538 |
lemma Step1_2_1: "|- [HNext rmhist p]_(c p,r p,m p, rmhist!p) & ImpNext p |
|
539 |
& ~unchanged (e p, c p, r p, m p, rmhist!p) & $S1 rmhist p |
|
540 |
--> (S2 rmhist p)$ & ENext p & unchanged (c p, r p, m p)" |
|
541 |
apply (tactic {* action_simp_tac (simpset() addsimps [thm "ImpNext_def"]) [] |
|
542 |
(map temp_elim [thm "S1ClerkUnch", thm "S1RPCUnch", thm "S1MemUnch", thm "S1Hist"]) 1 *}) |
|
543 |
apply (tactic {* auto_tac (MI_fast_css addSIs2 [temp_use (thm "S1Env")]) *}) |
|
544 |
done |
|
545 |
||
546 |
lemma Step1_2_2: "|- [HNext rmhist p]_(c p,r p,m p, rmhist!p) & ImpNext p |
|
547 |
& ~unchanged (e p, c p, r p, m p, rmhist!p) & $S2 rmhist p |
|
548 |
--> (S3 rmhist p)$ & MClkFwd memCh crCh cst p |
|
549 |
& unchanged (e p, r p, m p, rmhist!p)" |
|
550 |
apply (tactic {* action_simp_tac (simpset() addsimps [thm "ImpNext_def"]) [] |
|
551 |
(map temp_elim [thm "S2EnvUnch", thm "S2RPCUnch", thm "S2MemUnch", thm "S2Hist"]) 1 *}) |
|
552 |
apply (tactic {* auto_tac (MI_fast_css addSIs2 [temp_use (thm "S2Clerk"), |
|
553 |
temp_use (thm "S2Forward")]) *}) |
|
554 |
done |
|
555 |
||
556 |
lemma Step1_2_3: "|- [HNext rmhist p]_(c p,r p,m p, rmhist!p) & ImpNext p |
|
557 |
& ~unchanged (e p, c p, r p, m p, rmhist!p) & $S3 rmhist p |
|
558 |
--> ((S4 rmhist p)$ & RPCFwd crCh rmCh rst p & unchanged (e p, c p, m p, rmhist!p)) |
|
559 |
| ((S6 rmhist p)$ & RPCFail crCh rmCh rst p & unchanged (e p, c p, m p))" |
|
560 |
apply (tactic {* action_simp_tac (simpset() addsimps [thm "ImpNext_def"]) [] |
|
561 |
(map temp_elim [thm "S3EnvUnch", thm "S3ClerkUnch", thm "S3MemUnch"]) 1 *}) |
|
562 |
apply (tactic {* action_simp_tac (simpset()) [] |
|
563 |
(thm "squareE" :: map temp_elim [thm "S3RPC", thm "S3Forward", thm "S3Fail"]) 1 *}) |
|
564 |
apply (auto dest!: S3Hist [temp_use]) |
|
565 |
done |
|
566 |
||
567 |
lemma Step1_2_4: "|- [HNext rmhist p]_(c p,r p,m p, rmhist!p) & ImpNext p |
|
568 |
& ~unchanged (e p, c p, r p, m p, rmhist!p) |
|
569 |
& $S4 rmhist p & (!l. $(MemInv mm l)) |
|
570 |
--> ((S4 rmhist p)$ & Read rmCh mm ires p & unchanged (e p, c p, r p, rmhist!p)) |
|
571 |
| ((S4 rmhist p)$ & (? l. Write rmCh mm ires p l) & unchanged (e p, c p, r p, rmhist!p)) |
|
572 |
| ((S5 rmhist p)$ & MemReturn rmCh ires p & unchanged (e p, c p, r p))" |
|
573 |
apply (tactic {* action_simp_tac (simpset() addsimps [thm "ImpNext_def"]) [] |
|
574 |
(map temp_elim [thm "S4EnvUnch", thm "S4ClerkUnch", thm "S4RPCUnch"]) 1 *}) |
|
575 |
apply (tactic {* action_simp_tac (simpset() addsimps [thm "RNext_def"]) [] |
|
576 |
(thm "squareE" :: map temp_elim [thm "S4Read", thm "S4Write", thm "S4Return"]) 1 *}) |
|
577 |
apply (auto dest!: S4Hist [temp_use]) |
|
578 |
done |
|
579 |
||
580 |
lemma Step1_2_5: "|- [HNext rmhist p]_(c p,r p,m p, rmhist!p) & ImpNext p |
|
581 |
& ~unchanged (e p, c p, r p, m p, rmhist!p) & $S5 rmhist p |
|
582 |
--> ((S6 rmhist p)$ & RPCReply crCh rmCh rst p & unchanged (e p, c p, m p)) |
|
583 |
| ((S6 rmhist p)$ & RPCFail crCh rmCh rst p & unchanged (e p, c p, m p))" |
|
584 |
apply (tactic {* action_simp_tac (simpset() addsimps [thm "ImpNext_def"]) [] |
|
585 |
(map temp_elim [thm "S5EnvUnch", thm "S5ClerkUnch", thm "S5MemUnch", thm "S5Hist"]) 1 *}) |
|
586 |
apply (tactic {* action_simp_tac (simpset()) [] [thm "squareE", temp_elim (thm "S5RPC")] 1 *}) |
|
587 |
apply (tactic {* auto_tac (MI_fast_css addSDs2 |
|
588 |
[temp_use (thm "S5Reply"), temp_use (thm "S5Fail")]) *}) |
|
589 |
done |
|
590 |
||
591 |
lemma Step1_2_6: "|- [HNext rmhist p]_(c p,r p,m p, rmhist!p) & ImpNext p |
|
592 |
& ~unchanged (e p, c p, r p, m p, rmhist!p) & $S6 rmhist p |
|
593 |
--> ((S1 rmhist p)$ & MClkReply memCh crCh cst p & unchanged (e p, r p, m p)) |
|
594 |
| ((S3 rmhist p)$ & MClkRetry memCh crCh cst p & unchanged (e p,r p,m p,rmhist!p))" |
|
595 |
apply (tactic {* action_simp_tac (simpset() addsimps [thm "ImpNext_def"]) [] |
|
596 |
(map temp_elim [thm "S6EnvUnch", thm "S6RPCUnch", thm "S6MemUnch"]) 1 *}) |
|
597 |
apply (tactic {* action_simp_tac (simpset()) [] |
|
598 |
(thm "squareE" :: map temp_elim [thm "S6Clerk", thm "S6Retry", thm "S6Reply"]) 1 *}) |
|
599 |
apply (auto dest: S6Hist [temp_use]) |
|
600 |
done |
|
601 |
||
602 |
(* -------------------------------------------------------------------------- |
|
603 |
Step 1.3: S1 implies the barred initial condition. |
|
604 |
*) |
|
605 |
||
606 |
section "Initialization (Step 1.3)" |
|
607 |
||
608 |
lemma Step1_3: "|- S1 rmhist p --> PInit (resbar rmhist) p" |
|
609 |
by (tactic {* action_simp_tac (simpset () addsimps [thm "resbar_def", |
|
610 |
thm "PInit_def", thm "S_def", thm "S1_def"]) [] [] 1 *}) |
|
611 |
||
612 |
(* ---------------------------------------------------------------------- |
|
613 |
Step 1.4: Implementation's next-state relation simulates specification's |
|
614 |
next-state relation (with appropriate substitutions) |
|
615 |
*) |
|
616 |
||
617 |
section "Step simulation (Step 1.4)" |
|
618 |
||
619 |
lemma Step1_4_1: "|- ENext p & $S1 rmhist p & (S2 rmhist p)$ & unchanged (c p, r p, m p) |
|
620 |
--> unchanged (rtrner memCh!p, resbar rmhist!p)" |
|
621 |
by (tactic {* auto_tac (MI_fast_css addsimps2 [thm "c_def", thm "r_def", |
|
622 |
thm "m_def", thm "resbar_def"]) *}) |
|
623 |
||
624 |
lemma Step1_4_2: "|- MClkFwd memCh crCh cst p & $S2 rmhist p & (S3 rmhist p)$ |
|
625 |
& unchanged (e p, r p, m p, rmhist!p) |
|
626 |
--> unchanged (rtrner memCh!p, resbar rmhist!p)" |
|
627 |
by (tactic {* action_simp_tac |
|
628 |
(simpset() addsimps [thm "MClkFwd_def", thm "e_def", thm "r_def", thm "m_def", |
|
629 |
thm "resbar_def", thm "S_def", thm "S2_def", thm "S3_def"]) [] [] 1 *}) |
|
630 |
||
631 |
lemma Step1_4_3a: "|- RPCFwd crCh rmCh rst p & $S3 rmhist p & (S4 rmhist p)$ |
|
632 |
& unchanged (e p, c p, m p, rmhist!p) |
|
633 |
--> unchanged (rtrner memCh!p, resbar rmhist!p)" |
|
634 |
apply clarsimp |
|
635 |
apply (drule S3_excl [temp_use] S4_excl [temp_use])+ |
|
636 |
apply (tactic {* action_simp_tac (simpset () addsimps [thm "e_def", |
|
637 |
thm "c_def", thm "m_def", thm "resbar_def", thm "S_def", thm "S3_def"]) [] [] 1 *}) |
|
638 |
done |
|
639 |
||
640 |
lemma Step1_4_3b: "|- RPCFail crCh rmCh rst p & $S3 rmhist p & (S6 rmhist p)$ |
|
641 |
& unchanged (e p, c p, m p) |
|
642 |
--> MemFail memCh (resbar rmhist) p" |
|
643 |
apply clarsimp |
|
644 |
apply (drule S6_excl [temp_use]) |
|
645 |
apply (auto simp: RPCFail_def MemFail_def e_def c_def m_def resbar_def) |
|
646 |
apply (force simp: S3_def S_def) |
|
647 |
apply (auto simp: Return_def) |
|
648 |
done |
|
649 |
||
650 |
lemma Step1_4_4a1: "|- $S4 rmhist p & (S4 rmhist p)$ & ReadInner rmCh mm ires p l |
|
651 |
& unchanged (e p, c p, r p, rmhist!p) & $MemInv mm l |
|
652 |
--> ReadInner memCh mm (resbar rmhist) p l" |
|
653 |
apply clarsimp |
|
654 |
apply (drule S4_excl [temp_use])+ |
|
655 |
apply (tactic {* action_simp_tac (simpset () addsimps [thm "ReadInner_def", |
|
656 |
thm "GoodRead_def", thm "BadRead_def", thm "e_def", thm "c_def", thm "m_def"]) [] [] 1 *}) |
|
657 |
apply (auto simp: resbar_def) |
|
658 |
apply (tactic {* ALLGOALS (action_simp_tac |
|
659 |
(simpset() addsimps [thm "RPCRelayArg_def", thm "MClkRelayArg_def", |
|
660 |
thm "S_def", thm "S4_def", thm "RdRequest_def", thm "MemInv_def"]) |
|
661 |
[] [thm "impE", thm "MemValNotAResultE"]) *}) |
|
662 |
done |
|
663 |
||
664 |
lemma Step1_4_4a: "|- Read rmCh mm ires p & $S4 rmhist p & (S4 rmhist p)$ |
|
665 |
& unchanged (e p, c p, r p, rmhist!p) & (!l. $(MemInv mm l)) |
|
666 |
--> Read memCh mm (resbar rmhist) p" |
|
667 |
by (force simp: Read_def elim!: Step1_4_4a1 [temp_use]) |
|
668 |
||
669 |
lemma Step1_4_4b1: "|- $S4 rmhist p & (S4 rmhist p)$ & WriteInner rmCh mm ires p l v |
|
670 |
& unchanged (e p, c p, r p, rmhist!p) |
|
671 |
--> WriteInner memCh mm (resbar rmhist) p l v" |
|
672 |
apply clarsimp |
|
673 |
apply (drule S4_excl [temp_use])+ |
|
674 |
apply (tactic {* action_simp_tac (simpset () addsimps |
|
675 |
[thm "WriteInner_def", thm "GoodWrite_def", thm "BadWrite_def", thm "e_def", |
|
676 |
thm "c_def", thm "m_def"]) [] [] 1 *}) |
|
677 |
apply (auto simp: resbar_def) |
|
678 |
apply (tactic {* ALLGOALS (action_simp_tac (simpset () addsimps |
|
679 |
[thm "RPCRelayArg_def", thm "MClkRelayArg_def", thm "S_def", |
|
680 |
thm "S4_def", thm "WrRequest_def"]) [] []) *}) |
|
681 |
done |
|
682 |
||
683 |
lemma Step1_4_4b: "|- Write rmCh mm ires p l & $S4 rmhist p & (S4 rmhist p)$ |
|
684 |
& unchanged (e p, c p, r p, rmhist!p) |
|
685 |
--> Write memCh mm (resbar rmhist) p l" |
|
686 |
by (force simp: Write_def elim!: Step1_4_4b1 [temp_use]) |
|
687 |
||
688 |
lemma Step1_4_4c: "|- MemReturn rmCh ires p & $S4 rmhist p & (S5 rmhist p)$ |
|
689 |
& unchanged (e p, c p, r p) |
|
690 |
--> unchanged (rtrner memCh!p, resbar rmhist!p)" |
|
691 |
apply (tactic {* action_simp_tac (simpset () addsimps [thm "e_def", |
|
692 |
thm "c_def", thm "r_def", thm "resbar_def"]) [] [] 1 *}) |
|
693 |
apply (drule S4_excl [temp_use] S5_excl [temp_use])+ |
|
694 |
apply (tactic {* auto_tac (MI_fast_css addsimps2 [thm "MemReturn_def", thm "Return_def"]) *}) |
|
695 |
done |
|
696 |
||
697 |
lemma Step1_4_5a: "|- RPCReply crCh rmCh rst p & $S5 rmhist p & (S6 rmhist p)$ |
|
698 |
& unchanged (e p, c p, m p) |
|
699 |
--> unchanged (rtrner memCh!p, resbar rmhist!p)" |
|
700 |
apply clarsimp |
|
701 |
apply (drule S5_excl [temp_use] S6_excl [temp_use])+ |
|
702 |
apply (auto simp: e_def c_def m_def resbar_def) |
|
703 |
apply (auto simp: RPCReply_def Return_def S5_def S_def dest!: MVOKBAnotRF [temp_use]) |
|
704 |
done |
|
705 |
||
706 |
lemma Step1_4_5b: "|- RPCFail crCh rmCh rst p & $S5 rmhist p & (S6 rmhist p)$ |
|
707 |
& unchanged (e p, c p, m p) |
|
708 |
--> MemFail memCh (resbar rmhist) p" |
|
709 |
apply clarsimp |
|
710 |
apply (drule S6_excl [temp_use]) |
|
711 |
apply (auto simp: e_def c_def m_def RPCFail_def Return_def MemFail_def resbar_def) |
|
712 |
apply (auto simp: S5_def S_def) |
|
713 |
done |
|
714 |
||
715 |
lemma Step1_4_6a: "|- MClkReply memCh crCh cst p & $S6 rmhist p & (S1 rmhist p)$ |
|
716 |
& unchanged (e p, r p, m p) |
|
717 |
--> MemReturn memCh (resbar rmhist) p" |
|
718 |
apply clarsimp |
|
719 |
apply (drule S6_excl [temp_use])+ |
|
720 |
apply (tactic {* action_simp_tac (simpset () addsimps [thm "e_def", |
|
721 |
thm "r_def", thm "m_def", thm "MClkReply_def", thm "MemReturn_def", |
|
722 |
thm "Return_def", thm "resbar_def"]) [] [] 1 *}) |
|
723 |
apply simp_all (* simplify if-then-else *) |
|
724 |
apply (tactic {* ALLGOALS (action_simp_tac (simpset () addsimps |
|
725 |
[thm "MClkReplyVal_def", thm "S6_def", thm "S_def"]) [] [thm "MVOKBARFnotNR"]) *}) |
|
726 |
done |
|
727 |
||
728 |
lemma Step1_4_6b: "|- MClkRetry memCh crCh cst p & $S6 rmhist p & (S3 rmhist p)$ |
|
729 |
& unchanged (e p, r p, m p, rmhist!p) |
|
730 |
--> MemFail memCh (resbar rmhist) p" |
|
731 |
apply clarsimp |
|
732 |
apply (drule S3_excl [temp_use])+ |
|
733 |
apply (tactic {* action_simp_tac (simpset () addsimps [thm "e_def", thm "r_def", |
|
734 |
thm "m_def", thm "MClkRetry_def", thm "MemFail_def", thm "resbar_def"]) [] [] 1 *}) |
|
735 |
apply (auto simp: S6_def S_def) |
|
736 |
done |
|
737 |
||
738 |
lemma S_lemma: "|- unchanged (e p, c p, r p, m p, rmhist!p) |
|
739 |
--> unchanged (S rmhist ec cc rc cs rs hs1 hs2 p)" |
|
740 |
by (auto simp: e_def c_def r_def m_def caller_def rtrner_def S_def Calling_def) |
|
741 |
||
742 |
lemma Step1_4_7H: "|- unchanged (e p, c p, r p, m p, rmhist!p) |
|
743 |
--> unchanged (rtrner memCh!p, S1 rmhist p, S2 rmhist p, S3 rmhist p, |
|
744 |
S4 rmhist p, S5 rmhist p, S6 rmhist p)" |
|
745 |
apply clarsimp |
|
746 |
apply (rule conjI) |
|
747 |
apply (force simp: c_def) |
|
748 |
apply (force simp: S1_def S2_def S3_def S4_def S5_def S6_def intro!: S_lemma [temp_use]) |
|
749 |
done |
|
750 |
||
751 |
lemma Step1_4_7: "|- unchanged (e p, c p, r p, m p, rmhist!p) |
|
752 |
--> unchanged (rtrner memCh!p, resbar rmhist!p, S1 rmhist p, S2 rmhist p, |
|
753 |
S3 rmhist p, S4 rmhist p, S5 rmhist p, S6 rmhist p)" |
|
754 |
apply (rule actionI) |
|
755 |
apply (unfold action_rews) |
|
756 |
apply (rule impI) |
|
757 |
apply (frule Step1_4_7H [temp_use]) |
|
758 |
apply (auto simp: e_def c_def r_def m_def rtrner_def resbar_def) |
|
759 |
done |
|
760 |
||
761 |
(* Frequently needed abbreviation: distinguish between idling and non-idling |
|
762 |
steps of the implementation, and try to solve the idling case by simplification |
|
763 |
*) |
|
764 |
ML {* |
|
765 |
local |
|
766 |
val actionI = thm "actionI"; |
|
767 |
val action_rews = thms "action_rews"; |
|
768 |
val Step1_4_7 = thm "Step1_4_7"; |
|
769 |
in |
|
770 |
fun split_idle_tac simps i = |
|
771 |
EVERY [TRY (rtac actionI i), |
|
772 |
case_tac "(s,t) |= unchanged (e p, c p, r p, m p, rmhist!p)" i, |
|
773 |
rewrite_goals_tac action_rews, |
|
774 |
forward_tac [temp_use Step1_4_7] i, |
|
775 |
asm_full_simp_tac (simpset() addsimps simps) i |
|
776 |
] |
|
777 |
end |
|
778 |
*} |
|
779 |
(* ---------------------------------------------------------------------- |
|
780 |
Combine steps 1.2 and 1.4 to prove that the implementation satisfies |
|
781 |
the specification's next-state relation. |
|
782 |
*) |
|
783 |
||
784 |
(* Steps that leave all variables unchanged are safe, so I may assume |
|
785 |
that some variable changes in the proof that a step is safe. *) |
|
786 |
lemma unchanged_safe: "|- (~unchanged (e p, c p, r p, m p, rmhist!p) |
|
787 |
--> [UNext memCh mm (resbar rmhist) p]_(rtrner memCh!p, resbar rmhist!p)) |
|
788 |
--> [UNext memCh mm (resbar rmhist) p]_(rtrner memCh!p, resbar rmhist!p)" |
|
789 |
apply (tactic {* split_idle_tac [thm "square_def"] 1 *}) |
|
790 |
apply force |
|
791 |
done |
|
792 |
(* turn into (unsafe, looping!) introduction rule *) |
|
793 |
lemmas unchanged_safeI = impI [THEN unchanged_safe [action_use], standard] |
|
794 |
||
795 |
lemma S1safe: "|- $S1 rmhist p & ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p) |
|
796 |
--> [UNext memCh mm (resbar rmhist) p]_(rtrner memCh!p, resbar rmhist!p)" |
|
797 |
apply clarsimp |
|
798 |
apply (rule unchanged_safeI) |
|
799 |
apply (rule idle_squareI) |
|
800 |
apply (auto dest!: Step1_2_1 [temp_use] Step1_4_1 [temp_use]) |
|
801 |
done |
|
802 |
||
803 |
lemma S2safe: "|- $S2 rmhist p & ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p) |
|
804 |
--> [UNext memCh mm (resbar rmhist) p]_(rtrner memCh!p, resbar rmhist!p)" |
|
805 |
apply clarsimp |
|
806 |
apply (rule unchanged_safeI) |
|
807 |
apply (rule idle_squareI) |
|
808 |
apply (auto dest!: Step1_2_2 [temp_use] Step1_4_2 [temp_use]) |
|
809 |
done |
|
810 |
||
811 |
lemma S3safe: "|- $S3 rmhist p & ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p) |
|
812 |
--> [UNext memCh mm (resbar rmhist) p]_(rtrner memCh!p, resbar rmhist!p)" |
|
813 |
apply clarsimp |
|
814 |
apply (rule unchanged_safeI) |
|
815 |
apply (auto dest!: Step1_2_3 [temp_use]) |
|
816 |
apply (auto simp: square_def UNext_def dest!: Step1_4_3a [temp_use] Step1_4_3b [temp_use]) |
|
817 |
done |
|
818 |
||
819 |
lemma S4safe: "|- $S4 rmhist p & ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p) |
|
820 |
& (!l. $(MemInv mm l)) |
|
821 |
--> [UNext memCh mm (resbar rmhist) p]_(rtrner memCh!p, resbar rmhist!p)" |
|
822 |
apply clarsimp |
|
823 |
apply (rule unchanged_safeI) |
|
824 |
apply (auto dest!: Step1_2_4 [temp_use]) |
|
825 |
apply (auto simp: square_def UNext_def RNext_def |
|
826 |
dest!: Step1_4_4a [temp_use] Step1_4_4b [temp_use] Step1_4_4c [temp_use]) |
|
827 |
done |
|
828 |
||
829 |
lemma S5safe: "|- $S5 rmhist p & ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p) |
|
830 |
--> [UNext memCh mm (resbar rmhist) p]_(rtrner memCh!p, resbar rmhist!p)" |
|
831 |
apply clarsimp |
|
832 |
apply (rule unchanged_safeI) |
|
833 |
apply (auto dest!: Step1_2_5 [temp_use]) |
|
834 |
apply (auto simp: square_def UNext_def dest!: Step1_4_5a [temp_use] Step1_4_5b [temp_use]) |
|
835 |
done |
|
836 |
||
837 |
lemma S6safe: "|- $S6 rmhist p & ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p) |
|
838 |
--> [UNext memCh mm (resbar rmhist) p]_(rtrner memCh!p, resbar rmhist!p)" |
|
839 |
apply clarsimp |
|
840 |
apply (rule unchanged_safeI) |
|
841 |
apply (auto dest!: Step1_2_6 [temp_use]) |
|
842 |
apply (auto simp: square_def UNext_def RNext_def |
|
843 |
dest!: Step1_4_6a [temp_use] Step1_4_6b [temp_use]) |
|
844 |
done |
|
845 |
||
846 |
(* ---------------------------------------------------------------------- |
|
847 |
Step 1.5: Temporal refinement proof, based on previous steps. |
|
848 |
*) |
|
849 |
||
850 |
section "The liveness part" |
|
851 |
||
852 |
(* Liveness assertions for the different implementation states, based on the |
|
853 |
fairness conditions. Prove subgoals of WF1 / SF1 rules as separate lemmas |
|
854 |
for readability. Reuse action proofs from safety part. |
|
855 |
*) |
|
856 |
||
857 |
(* ------------------------------ State S1 ------------------------------ *) |
|
858 |
||
859 |
lemma S1_successors: "|- $S1 rmhist p & ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p) |
|
860 |
--> (S1 rmhist p)$ | (S2 rmhist p)$" |
|
861 |
apply (tactic "split_idle_tac [] 1") |
|
862 |
apply (auto dest!: Step1_2_1 [temp_use]) |
|
863 |
done |
|
864 |
||
865 |
(* Show that the implementation can satisfy the high-level fairness requirements |
|
866 |
by entering the state S1 infinitely often. |
|
867 |
*) |
|
868 |
||
869 |
lemma S1_RNextdisabled: "|- S1 rmhist p --> |
|
870 |
~Enabled (<RNext memCh mm (resbar rmhist) p>_(rtrner memCh!p, resbar rmhist!p))" |
|
871 |
apply (tactic {* action_simp_tac (simpset () addsimps [thm "angle_def", |
|
872 |
thm "S_def", thm "S1_def"]) [notI] [thm "enabledE", temp_elim (thm "Memoryidle")] 1 *}) |
|
873 |
apply force |
|
874 |
done |
|
875 |
||
876 |
lemma S1_Returndisabled: "|- S1 rmhist p --> |
|
877 |
~Enabled (<MemReturn memCh (resbar rmhist) p>_(rtrner memCh!p, resbar rmhist!p))" |
|
878 |
by (tactic {* action_simp_tac (simpset () addsimps [thm "angle_def", thm "MemReturn_def", |
|
879 |
thm "Return_def", thm "S_def", thm "S1_def"]) [notI] [thm "enabledE"] 1 *}) |
|
880 |
||
881 |
lemma RNext_fair: "|- []<>S1 rmhist p |
|
882 |
--> WF(RNext memCh mm (resbar rmhist) p)_(rtrner memCh!p, resbar rmhist!p)" |
|
883 |
by (auto simp: WF_alt [try_rewrite] intro!: S1_RNextdisabled [temp_use] |
|
884 |
elim!: STL4E [temp_use] DmdImplE [temp_use]) |
|
885 |
||
886 |
lemma Return_fair: "|- []<>S1 rmhist p |
|
887 |
--> WF(MemReturn memCh (resbar rmhist) p)_(rtrner memCh!p, resbar rmhist!p)" |
|
888 |
by (auto simp: WF_alt [try_rewrite] |
|
889 |
intro!: S1_Returndisabled [temp_use] elim!: STL4E [temp_use] DmdImplE [temp_use]) |
|
890 |
||
891 |
(* ------------------------------ State S2 ------------------------------ *) |
|
892 |
||
893 |
lemma S2_successors: "|- $S2 rmhist p & ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p) |
|
894 |
--> (S2 rmhist p)$ | (S3 rmhist p)$" |
|
895 |
apply (tactic "split_idle_tac [] 1") |
|
896 |
apply (auto dest!: Step1_2_2 [temp_use]) |
|
897 |
done |
|
898 |
||
899 |
lemma S2MClkFwd_successors: "|- ($S2 rmhist p & ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p)) |
|
900 |
& <MClkFwd memCh crCh cst p>_(c p) |
|
901 |
--> (S3 rmhist p)$" |
|
902 |
by (auto simp: angle_def dest!: Step1_2_2 [temp_use]) |
|
903 |
||
904 |
lemma S2MClkFwd_enabled: "|- $S2 rmhist p & ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p) |
|
905 |
--> $Enabled (<MClkFwd memCh crCh cst p>_(c p))" |
|
906 |
apply (auto simp: c_def intro!: MClkFwd_ch_enabled [temp_use] MClkFwd_enabled [temp_use]) |
|
907 |
apply (cut_tac MI_base) |
|
908 |
apply (blast dest: base_pair) |
|
909 |
apply (simp_all add: S_def S2_def) |
|
910 |
done |
|
911 |
||
912 |
lemma S2_live: "|- [](ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p)) |
|
913 |
& WF(MClkFwd memCh crCh cst p)_(c p) |
|
914 |
--> (S2 rmhist p ~> S3 rmhist p)" |
|
915 |
by (rule WF1 S2_successors S2MClkFwd_successors S2MClkFwd_enabled)+ |
|
916 |
||
917 |
(* ------------------------------ State S3 ------------------------------ *) |
|
918 |
||
919 |
lemma S3_successors: "|- $S3 rmhist p & ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p) |
|
920 |
--> (S3 rmhist p)$ | (S4 rmhist p | S6 rmhist p)$" |
|
921 |
apply (tactic "split_idle_tac [] 1") |
|
922 |
apply (auto dest!: Step1_2_3 [temp_use]) |
|
923 |
done |
|
924 |
||
925 |
lemma S3RPC_successors: "|- ($S3 rmhist p & ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p)) |
|
926 |
& <RPCNext crCh rmCh rst p>_(r p) |
|
927 |
--> (S4 rmhist p | S6 rmhist p)$" |
|
928 |
apply (auto simp: angle_def dest!: Step1_2_3 [temp_use]) |
|
929 |
done |
|
930 |
||
931 |
lemma S3RPC_enabled: "|- $S3 rmhist p & ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p) |
|
932 |
--> $Enabled (<RPCNext crCh rmCh rst p>_(r p))" |
|
933 |
apply (auto simp: r_def intro!: RPCFail_Next_enabled [temp_use] RPCFail_enabled [temp_use]) |
|
934 |
apply (cut_tac MI_base) |
|
935 |
apply (blast dest: base_pair) |
|
936 |
apply (simp_all add: S_def S3_def) |
|
937 |
done |
|
938 |
||
939 |
lemma S3_live: "|- [](ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p)) |
|
940 |
& WF(RPCNext crCh rmCh rst p)_(r p) |
|
941 |
--> (S3 rmhist p ~> S4 rmhist p | S6 rmhist p)" |
|
942 |
by (rule WF1 S3_successors S3RPC_successors S3RPC_enabled)+ |
|
943 |
||
944 |
(* ------------- State S4 -------------------------------------------------- *) |
|
945 |
||
946 |
lemma S4_successors: "|- $S4 rmhist p & ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p) |
|
947 |
& (ALL l. $MemInv mm l) |
|
948 |
--> (S4 rmhist p)$ | (S5 rmhist p)$" |
|
949 |
apply (tactic "split_idle_tac [] 1") |
|
950 |
apply (auto dest!: Step1_2_4 [temp_use]) |
|
951 |
done |
|
952 |
||
953 |
(* --------- State S4a: S4 /\ (ires p = NotAResult) ------------------------ *) |
|
954 |
||
955 |
lemma S4a_successors: "|- $(S4 rmhist p & ires!p = #NotAResult) |
|
956 |
& ImpNext p & [HNext rmhist p]_(c p,r p,m p,rmhist!p) & (ALL l. $MemInv mm l) |
|
957 |
--> (S4 rmhist p & ires!p = #NotAResult)$ |
|
958 |
| ((S4 rmhist p & ires!p ~= #NotAResult) | S5 rmhist p)$" |
|
959 |
apply (tactic {* split_idle_tac [thm "m_def"] 1 *}) |
|
960 |
apply (auto dest!: Step1_2_4 [temp_use]) |
|
961 |
done |
|
962 |
||
963 |
lemma S4aRNext_successors: "|- ($(S4 rmhist p & ires!p = #NotAResult) |
|
964 |
& ImpNext p & [HNext rmhist p]_(c p,r p,m p,rmhist!p) & (ALL l. $MemInv mm l)) |
|
965 |
& <RNext rmCh mm ires p>_(m p) |
|
966 |
--> ((S4 rmhist p & ires!p ~= #NotAResult) | S5 rmhist p)$" |
|
967 |
by (auto simp: angle_def |
|
968 |
dest!: Step1_2_4 [temp_use] ReadResult [temp_use] WriteResult [temp_use]) |
|
969 |
||
970 |
lemma S4aRNext_enabled: "|- $(S4 rmhist p & ires!p = #NotAResult) |
|
971 |
& ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p) & (ALL l. $MemInv mm l) |
|
972 |
--> $Enabled (<RNext rmCh mm ires p>_(m p))" |
|
973 |
apply (auto simp: m_def intro!: RNext_enabled [temp_use]) |
|
974 |
apply (cut_tac MI_base) |
|
975 |
apply (blast dest: base_pair) |
|
976 |
apply (simp add: S_def S4_def) |
|
977 |
done |
|
978 |
||
979 |
lemma S4a_live: "|- [](ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p) |
|
980 |
& (ALL l. $MemInv mm l)) & WF(RNext rmCh mm ires p)_(m p) |
|
981 |
--> (S4 rmhist p & ires!p = #NotAResult |
|
982 |
~> (S4 rmhist p & ires!p ~= #NotAResult) | S5 rmhist p)" |
|
983 |
by (rule WF1 S4a_successors S4aRNext_successors S4aRNext_enabled)+ |
|
984 |
||
985 |
(* ---------- State S4b: S4 /\ (ires p # NotAResult) --------------------------- *) |
|
986 |
||
987 |
lemma S4b_successors: "|- $(S4 rmhist p & ires!p ~= #NotAResult) |
|
988 |
& ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p) & (ALL l. $MemInv mm l) |
|
989 |
--> (S4 rmhist p & ires!p ~= #NotAResult)$ | (S5 rmhist p)$" |
|
990 |
apply (tactic {* split_idle_tac [thm "m_def"] 1 *}) |
|
991 |
apply (auto dest!: WriteResult [temp_use] Step1_2_4 [temp_use] ReadResult [temp_use]) |
|
992 |
done |
|
993 |
||
994 |
lemma S4bReturn_successors: "|- ($(S4 rmhist p & ires!p ~= #NotAResult) |
|
995 |
& ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p) |
|
996 |
& (ALL l. $MemInv mm l)) & <MemReturn rmCh ires p>_(m p) |
|
997 |
--> (S5 rmhist p)$" |
|
998 |
by (force simp: angle_def dest!: Step1_2_4 [temp_use] dest: ReturnNotReadWrite [temp_use]) |
|
999 |
||
1000 |
lemma S4bReturn_enabled: "|- $(S4 rmhist p & ires!p ~= #NotAResult) |
|
1001 |
& ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p) |
|
1002 |
& (ALL l. $MemInv mm l) |
|
1003 |
--> $Enabled (<MemReturn rmCh ires p>_(m p))" |
|
1004 |
apply (auto simp: m_def intro!: MemReturn_enabled [temp_use]) |
|
1005 |
apply (cut_tac MI_base) |
|
1006 |
apply (blast dest: base_pair) |
|
1007 |
apply (simp add: S_def S4_def) |
|
1008 |
done |
|
1009 |
||
1010 |
lemma S4b_live: "|- [](ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p) & (!l. $MemInv mm l)) |
|
1011 |
& WF(MemReturn rmCh ires p)_(m p) |
|
1012 |
--> (S4 rmhist p & ires!p ~= #NotAResult ~> S5 rmhist p)" |
|
1013 |
by (rule WF1 S4b_successors S4bReturn_successors S4bReturn_enabled)+ |
|
1014 |
||
1015 |
(* ------------------------------ State S5 ------------------------------ *) |
|
1016 |
||
1017 |
lemma S5_successors: "|- $S5 rmhist p & ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p) |
|
1018 |
--> (S5 rmhist p)$ | (S6 rmhist p)$" |
|
1019 |
apply (tactic "split_idle_tac [] 1") |
|
1020 |
apply (auto dest!: Step1_2_5 [temp_use]) |
|
1021 |
done |
|
1022 |
||
1023 |
lemma S5RPC_successors: "|- ($S5 rmhist p & ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p)) |
|
1024 |
& <RPCNext crCh rmCh rst p>_(r p) |
|
1025 |
--> (S6 rmhist p)$" |
|
1026 |
by (auto simp: angle_def dest!: Step1_2_5 [temp_use]) |
|
1027 |
||
1028 |
lemma S5RPC_enabled: "|- $S5 rmhist p & ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p) |
|
1029 |
--> $Enabled (<RPCNext crCh rmCh rst p>_(r p))" |
|
1030 |
apply (auto simp: r_def intro!: RPCFail_Next_enabled [temp_use] RPCFail_enabled [temp_use]) |
|
1031 |
apply (cut_tac MI_base) |
|
1032 |
apply (blast dest: base_pair) |
|
1033 |
apply (simp_all add: S_def S5_def) |
|
1034 |
done |
|
1035 |
||
1036 |
lemma S5_live: "|- [](ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p)) |
|
1037 |
& WF(RPCNext crCh rmCh rst p)_(r p) |
|
1038 |
--> (S5 rmhist p ~> S6 rmhist p)" |
|
1039 |
by (rule WF1 S5_successors S5RPC_successors S5RPC_enabled)+ |
|
1040 |
||
1041 |
(* ------------------------------ State S6 ------------------------------ *) |
|
1042 |
||
1043 |
lemma S6_successors: "|- $S6 rmhist p & ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p) |
|
1044 |
--> (S1 rmhist p)$ | (S3 rmhist p)$ | (S6 rmhist p)$" |
|
1045 |
apply (tactic "split_idle_tac [] 1") |
|
1046 |
apply (auto dest!: Step1_2_6 [temp_use]) |
|
1047 |
done |
|
1048 |
||
1049 |
lemma S6MClkReply_successors: |
|
1050 |
"|- ($S6 rmhist p & ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p)) |
|
1051 |
& <MClkReply memCh crCh cst p>_(c p) |
|
1052 |
--> (S1 rmhist p)$" |
|
1053 |
by (auto simp: angle_def dest!: Step1_2_6 [temp_use] MClkReplyNotRetry [temp_use]) |
|
1054 |
||
1055 |
lemma MClkReplyS6: |
|
1056 |
"|- $ImpInv rmhist p & <MClkReply memCh crCh cst p>_(c p) --> $S6 rmhist p" |
|
1057 |
by (tactic {* action_simp_tac (simpset () addsimps [thm "angle_def", |
|
1058 |
thm "MClkReply_def", thm "Return_def", thm "ImpInv_def", thm "S_def", |
|
1059 |
thm "S1_def", thm "S2_def", thm "S3_def", thm "S4_def", thm "S5_def"]) [] [] 1 *}) |
|
1060 |
||
1061 |
lemma S6MClkReply_enabled: "|- S6 rmhist p --> Enabled (<MClkReply memCh crCh cst p>_(c p))" |
|
1062 |
apply (auto simp: c_def intro!: MClkReply_enabled [temp_use]) |
|
1063 |
apply (cut_tac MI_base) |
|
1064 |
apply (blast dest: base_pair) |
|
1065 |
apply (tactic {* ALLGOALS (action_simp_tac (simpset () |
|
1066 |
addsimps [thm "S_def", thm "S6_def"]) [] []) *}) |
|
1067 |
done |
|
1068 |
||
1069 |
lemma S6_live: "|- [](ImpNext p & [HNext rmhist p]_(c p,r p,m p, rmhist!p) & $(ImpInv rmhist p)) |
|
1070 |
& SF(MClkReply memCh crCh cst p)_(c p) & []<>(S6 rmhist p) |
|
1071 |
--> []<>(S1 rmhist p)" |
|
1072 |
apply clarsimp |
|
1073 |
apply (subgoal_tac "sigma |= []<> (<MClkReply memCh crCh cst p>_ (c p))") |
|
1074 |
apply (erule InfiniteEnsures) |
|
1075 |
apply assumption |
|
1076 |
apply (tactic {* action_simp_tac (simpset()) [] |
|
1077 |
(map temp_elim [thm "MClkReplyS6", thm "S6MClkReply_successors"]) 1 *}) |
|
1078 |
apply (auto simp: SF_def) |
|
1079 |
apply (erule contrapos_np) |
|
1080 |
apply (auto intro!: S6MClkReply_enabled [temp_use] elim!: STL4E [temp_use] DmdImplE [temp_use]) |
|
1081 |
done |
|
1082 |
||
1083 |
(* --------------- aggregate leadsto properties----------------------------- *) |
|
1084 |
||
1085 |
lemma S5S6LeadstoS6: "sigma |= S5 rmhist p ~> S6 rmhist p |
|
1086 |
==> sigma |= (S5 rmhist p | S6 rmhist p) ~> S6 rmhist p" |
|
1087 |
by (auto intro!: LatticeDisjunctionIntro [temp_use] LatticeReflexivity [temp_use]) |
|
1088 |
||
1089 |
lemma S4bS5S6LeadstoS6: "[| sigma |= S4 rmhist p & ires!p ~= #NotAResult ~> S5 rmhist p; |
|
1090 |
sigma |= S5 rmhist p ~> S6 rmhist p |] |
|
1091 |
==> sigma |= (S4 rmhist p & ires!p ~= #NotAResult) | S5 rmhist p | S6 rmhist p |
|
1092 |
~> S6 rmhist p" |
|
1093 |
by (auto intro!: LatticeDisjunctionIntro [temp_use] |
|
1094 |
S5S6LeadstoS6 [temp_use] intro: LatticeTransitivity [temp_use]) |
|
1095 |
||
1096 |
lemma S4S5S6LeadstoS6: "[| sigma |= S4 rmhist p & ires!p = #NotAResult |
|
1097 |
~> (S4 rmhist p & ires!p ~= #NotAResult) | S5 rmhist p; |
|
1098 |
sigma |= S4 rmhist p & ires!p ~= #NotAResult ~> S5 rmhist p; |
|
1099 |
sigma |= S5 rmhist p ~> S6 rmhist p |] |
|
1100 |
==> sigma |= S4 rmhist p | S5 rmhist p | S6 rmhist p ~> S6 rmhist p" |
|
1101 |
apply (subgoal_tac "sigma |= (S4 rmhist p & ires!p = #NotAResult) | |
|
1102 |
(S4 rmhist p & ires!p ~= #NotAResult) | S5 rmhist p | S6 rmhist p ~> S6 rmhist p") |
|
1103 |
apply (erule_tac G = "PRED ((S4 rmhist p & ires!p = #NotAResult) | |
|
1104 |
(S4 rmhist p & ires!p ~= #NotAResult) | S5 rmhist p | S6 rmhist p)" in |
|
1105 |
LatticeTransitivity [temp_use]) |
|
1106 |
apply (force simp: Init_defs intro!: ImplLeadsto_gen [temp_use] necT [temp_use]) |
|
1107 |
apply (rule LatticeDisjunctionIntro [temp_use]) |
|
1108 |
apply (erule LatticeTransitivity [temp_use]) |
|
1109 |
apply (erule LatticeTriangle2 [temp_use]) |
|
1110 |
apply assumption |
|
1111 |
apply (auto intro!: S4bS5S6LeadstoS6 [temp_use]) |
|
1112 |
done |
|
1113 |
||
1114 |
lemma S3S4S5S6LeadstoS6: "[| sigma |= S3 rmhist p ~> S4 rmhist p | S6 rmhist p; |
|
1115 |
sigma |= S4 rmhist p & ires!p = #NotAResult |
|
1116 |
~> (S4 rmhist p & ires!p ~= #NotAResult) | S5 rmhist p; |
|
1117 |
sigma |= S4 rmhist p & ires!p ~= #NotAResult ~> S5 rmhist p; |
|
1118 |
sigma |= S5 rmhist p ~> S6 rmhist p |] |
|
1119 |
==> sigma |= S3 rmhist p | S4 rmhist p | S5 rmhist p | S6 rmhist p ~> S6 rmhist p" |
|
1120 |
apply (rule LatticeDisjunctionIntro [temp_use]) |
|
1121 |
apply (erule LatticeTriangle2 [temp_use]) |
|
1122 |
apply (rule S4S5S6LeadstoS6 [THEN LatticeTransitivity [temp_use]]) |
|
1123 |
apply (auto intro!: S4S5S6LeadstoS6 [temp_use] necT [temp_use] |
|
1124 |
intro: ImplLeadsto_gen [temp_use] simp: Init_defs) |
|
1125 |
done |
|
1126 |
||
1127 |
lemma S2S3S4S5S6LeadstoS6: "[| sigma |= S2 rmhist p ~> S3 rmhist p; |
|
1128 |
sigma |= S3 rmhist p ~> S4 rmhist p | S6 rmhist p; |
|
1129 |
sigma |= S4 rmhist p & ires!p = #NotAResult |
|
1130 |
~> S4 rmhist p & ires!p ~= #NotAResult | S5 rmhist p; |
|
1131 |
sigma |= S4 rmhist p & ires!p ~= #NotAResult ~> S5 rmhist p; |
|
1132 |
sigma |= S5 rmhist p ~> S6 rmhist p |] |
|
1133 |
==> sigma |= S2 rmhist p | S3 rmhist p | S4 rmhist p | S5 rmhist p | S6 rmhist p |
|
1134 |
~> S6 rmhist p" |
|
1135 |
apply (rule LatticeDisjunctionIntro [temp_use]) |
|
1136 |
apply (rule LatticeTransitivity [temp_use]) |
|
1137 |
prefer 2 apply assumption |
|
1138 |
apply (rule S3S4S5S6LeadstoS6 [THEN LatticeTransitivity [temp_use]]) |
|
1139 |
apply (auto intro!: S3S4S5S6LeadstoS6 [temp_use] necT [temp_use] |
|
1140 |
intro: ImplLeadsto_gen [temp_use] simp: Init_defs) |
|
1141 |
done |
|
1142 |
||
1143 |
lemma NotS1LeadstoS6: "[| sigma |= []ImpInv rmhist p; |
|
1144 |
sigma |= S2 rmhist p ~> S3 rmhist p; |
|
1145 |
sigma |= S3 rmhist p ~> S4 rmhist p | S6 rmhist p; |
|
1146 |
sigma |= S4 rmhist p & ires!p = #NotAResult |
|
1147 |
~> S4 rmhist p & ires!p ~= #NotAResult | S5 rmhist p; |
|
1148 |
sigma |= S4 rmhist p & ires!p ~= #NotAResult ~> S5 rmhist p; |
|
1149 |
sigma |= S5 rmhist p ~> S6 rmhist p |] |
|
1150 |
==> sigma |= ~S1 rmhist p ~> S6 rmhist p" |
|
1151 |
apply (rule S2S3S4S5S6LeadstoS6 [THEN LatticeTransitivity [temp_use]]) |
|
1152 |
apply assumption+ |
|
1153 |
apply (erule INV_leadsto [temp_use]) |
|
1154 |
apply (rule ImplLeadsto_gen [temp_use]) |
|
1155 |
apply (rule necT [temp_use]) |
|
1156 |
apply (auto simp: ImpInv_def Init_defs intro!: necT [temp_use]) |
|
1157 |
done |
|
1158 |
||
1159 |
lemma S1Infinite: "[| sigma |= ~S1 rmhist p ~> S6 rmhist p; |
|
1160 |
sigma |= []<>S6 rmhist p --> []<>S1 rmhist p |] |
|
1161 |
==> sigma |= []<>S1 rmhist p" |
|
1162 |
apply (rule classical) |
|
1163 |
apply (tactic {* asm_lr_simp_tac (simpset() addsimps |
|
1164 |
[temp_use (thm "NotBox"), temp_rewrite (thm "NotDmd")]) 1 *}) |
|
1165 |
apply (auto elim!: leadsto_infinite [temp_use] mp dest!: DBImplBD [temp_use]) |
|
1166 |
done |
|
1167 |
||
1168 |
section "Refinement proof (step 1.5)" |
|
1169 |
||
1170 |
(* Prove invariants of the implementation: |
|
1171 |
a. memory invariant |
|
1172 |
b. "implementation invariant": always in states S1,...,S6 |
|
1173 |
*) |
|
1174 |
lemma Step1_5_1a: "|- IPImp p --> (ALL l. []$MemInv mm l)" |
|
1175 |
by (auto simp: IPImp_def box_stp_act [temp_use] intro!: MemoryInvariantAll [temp_use]) |
|
1176 |
||
1177 |
lemma Step1_5_1b: "|- Init(ImpInit p & HInit rmhist p) & [](ImpNext p) |
|
1178 |
& [][HNext rmhist p]_(c p, r p, m p, rmhist!p) & [](ALL l. $MemInv mm l) |
|
1179 |
--> []ImpInv rmhist p" |
|
1180 |
apply (tactic "inv_tac MI_css 1") |
|
1181 |
apply (auto simp: Init_def ImpInv_def box_stp_act [temp_use] |
|
1182 |
dest!: Step1_1 [temp_use] dest: S1_successors [temp_use] S2_successors [temp_use] |
|
1183 |
S3_successors [temp_use] S4_successors [temp_use] S5_successors [temp_use] |
|
1184 |
S6_successors [temp_use]) |
|
1185 |
done |
|
1186 |
||
1187 |
(*** Initialization ***) |
|
1188 |
lemma Step1_5_2a: "|- Init(ImpInit p & HInit rmhist p) --> Init(PInit (resbar rmhist) p)" |
|
1189 |
by (auto simp: Init_def intro!: Step1_1 [temp_use] Step1_3 [temp_use]) |
|
1190 |
||
1191 |
(*** step simulation ***) |
|
1192 |
lemma Step1_5_2b: "|- [](ImpNext p & [HNext rmhist p]_(c p, r p, m p, rmhist!p) |
|
1193 |
& $ImpInv rmhist p & (!l. $MemInv mm l)) |
|
1194 |
--> [][UNext memCh mm (resbar rmhist) p]_(rtrner memCh!p, resbar rmhist!p)" |
|
1195 |
by (auto simp: ImpInv_def elim!: STL4E [temp_use] |
|
1196 |
dest!: S1safe [temp_use] S2safe [temp_use] S3safe [temp_use] S4safe [temp_use] |
|
1197 |
S5safe [temp_use] S6safe [temp_use]) |
|
1198 |
||
1199 |
(*** Liveness ***) |
|
1200 |
lemma GoodImpl: "|- IPImp p & HistP rmhist p |
|
1201 |
--> Init(ImpInit p & HInit rmhist p) |
|
1202 |
& [](ImpNext p & [HNext rmhist p]_(c p, r p, m p, rmhist!p)) |
|
1203 |
& [](ALL l. $MemInv mm l) & []($ImpInv rmhist p) |
|
1204 |
& ImpLive p" |
|
1205 |
apply clarsimp |
|
1206 |
apply (subgoal_tac "sigma |= Init (ImpInit p & HInit rmhist p) & [] (ImpNext p) & |
|
1207 |
[][HNext rmhist p]_ (c p, r p, m p, rmhist!p) & [] (ALL l. $MemInv mm l)") |
|
1208 |
apply (auto simp: split_box_conj [try_rewrite] box_stp_act [try_rewrite] |
|
1209 |
dest!: Step1_5_1b [temp_use]) |
|
1210 |
apply (force simp: IPImp_def MClkIPSpec_def RPCIPSpec_def RPSpec_def |
|
1211 |
ImpLive_def c_def r_def m_def) |
|
1212 |
apply (force simp: IPImp_def MClkIPSpec_def RPCIPSpec_def RPSpec_def |
|
1213 |
HistP_def Init_def ImpInit_def) |
|
1214 |
apply (force simp: IPImp_def MClkIPSpec_def RPCIPSpec_def RPSpec_def |
|
1215 |
ImpNext_def c_def r_def m_def split_box_conj [temp_use]) |
|
1216 |
apply (force simp: HistP_def) |
|
1217 |
apply (force simp: allT [temp_use] dest!: Step1_5_1a [temp_use]) |
|
1218 |
done |
|
1219 |
||
1220 |
(* The implementation is infinitely often in state S1... *) |
|
1221 |
lemma Step1_5_3a: "|- [](ImpNext p & [HNext rmhist p]_(c p, r p, m p, rmhist!p)) |
|
1222 |
& [](ALL l. $MemInv mm l) |
|
1223 |
& []($ImpInv rmhist p) & ImpLive p |
|
1224 |
--> []<>S1 rmhist p" |
|
1225 |
apply (clarsimp simp: ImpLive_def) |
|
1226 |
apply (rule S1Infinite) |
|
1227 |
apply (force simp: split_box_conj [try_rewrite] box_stp_act [try_rewrite] |
|
1228 |
intro!: NotS1LeadstoS6 [temp_use] S2_live [temp_use] S3_live [temp_use] |
|
1229 |
S4a_live [temp_use] S4b_live [temp_use] S5_live [temp_use]) |
|
1230 |
apply (auto simp: split_box_conj [temp_use] intro!: S6_live [temp_use]) |
|
1231 |
done |
|
1232 |
||
1233 |
(* ... and therefore satisfies the fairness requirements of the specification *) |
|
1234 |
lemma Step1_5_3b: "|- [](ImpNext p & [HNext rmhist p]_(c p, r p, m p, rmhist!p)) |
|
1235 |
& [](ALL l. $MemInv mm l) & []($ImpInv rmhist p) & ImpLive p |
|
1236 |
--> WF(RNext memCh mm (resbar rmhist) p)_(rtrner memCh!p, resbar rmhist!p)" |
|
1237 |
by (auto intro!: RNext_fair [temp_use] Step1_5_3a [temp_use]) |
|
1238 |
||
1239 |
lemma Step1_5_3c: "|- [](ImpNext p & [HNext rmhist p]_(c p, r p, m p, rmhist!p)) |
|
1240 |
& [](ALL l. $MemInv mm l) & []($ImpInv rmhist p) & ImpLive p |
|
1241 |
--> WF(MemReturn memCh (resbar rmhist) p)_(rtrner memCh!p, resbar rmhist!p)" |
|
1242 |
by (auto intro!: Return_fair [temp_use] Step1_5_3a [temp_use]) |
|
1243 |
||
1244 |
(* QED step of step 1 *) |
|
1245 |
lemma Step1: "|- IPImp p & HistP rmhist p --> UPSpec memCh mm (resbar rmhist) p" |
|
1246 |
by (auto simp: UPSpec_def split_box_conj [temp_use] |
|
1247 |
dest!: GoodImpl [temp_use] intro!: Step1_5_2a [temp_use] Step1_5_2b [temp_use] |
|
1248 |
Step1_5_3b [temp_use] Step1_5_3c [temp_use]) |
|
1249 |
||
1250 |
(* ------------------------------ Step 2 ------------------------------ *) |
|
1251 |
section "Step 2" |
|
1252 |
||
1253 |
lemma Step2_2a: "|- Write rmCh mm ires p l & ImpNext p |
|
1254 |
& [HNext rmhist p]_(c p, r p, m p, rmhist!p) |
|
1255 |
& $ImpInv rmhist p |
|
1256 |
--> (S4 rmhist p)$ & unchanged (e p, c p, r p, rmhist!p)" |
|
1257 |
apply clarsimp |
|
1258 |
apply (drule WriteS4 [action_use]) |
|
1259 |
apply assumption |
|
1260 |
apply (tactic "split_idle_tac [] 1") |
|
1261 |
apply (auto simp: ImpNext_def dest!: S4EnvUnch [temp_use] S4ClerkUnch [temp_use] |
|
1262 |
S4RPCUnch [temp_use]) |
|
1263 |
apply (auto simp: square_def dest: S4Write [temp_use]) |
|
1264 |
done |
|
1265 |
||
1266 |
lemma Step2_2: "|- (ALL p. ImpNext p) |
|
1267 |
& (ALL p. [HNext rmhist p]_(c p, r p, m p, rmhist!p)) |
|
1268 |
& (ALL p. $ImpInv rmhist p) |
|
1269 |
& [EX q. Write rmCh mm ires q l]_(mm!l) |
|
1270 |
--> [EX q. Write memCh mm (resbar rmhist) q l]_(mm!l)" |
|
1271 |
apply (auto intro!: squareCI elim!: squareE) |
|
1272 |
apply (assumption | rule exI Step1_4_4b [action_use])+ |
|
1273 |
apply (force intro!: WriteS4 [temp_use]) |
|
1274 |
apply (auto dest!: Step2_2a [temp_use]) |
|
1275 |
done |
|
1276 |
||
1277 |
lemma Step2_lemma: "|- []( (ALL p. ImpNext p) |
|
1278 |
& (ALL p. [HNext rmhist p]_(c p, r p, m p, rmhist!p)) |
|
1279 |
& (ALL p. $ImpInv rmhist p) |
|
1280 |
& [EX q. Write rmCh mm ires q l]_(mm!l)) |
|
1281 |
--> [][EX q. Write memCh mm (resbar rmhist) q l]_(mm!l)" |
|
1282 |
by (force elim!: STL4E [temp_use] dest!: Step2_2 [temp_use]) |
|
1283 |
||
1284 |
lemma Step2: "|- #l : #MemLoc & (ALL p. IPImp p & HistP rmhist p) |
|
1285 |
--> MSpec memCh mm (resbar rmhist) l" |
|
1286 |
apply (auto simp: MSpec_def) |
|
1287 |
apply (force simp: IPImp_def MSpec_def) |
|
1288 |
apply (auto intro!: Step2_lemma [temp_use] simp: split_box_conj [temp_use] all_box [temp_use]) |
|
1289 |
prefer 4 |
|
1290 |
apply (force simp: IPImp_def MSpec_def) |
|
1291 |
apply (auto simp: split_box_conj [temp_use] elim!: allE dest!: GoodImpl [temp_use]) |
|
1292 |
done |
|
1293 |
||
1294 |
(* ----------------------------- Main theorem --------------------------------- *) |
|
1295 |
section "Memory implementation" |
|
1296 |
||
1297 |
(* The combination of a legal caller, the memory clerk, the RPC component, |
|
1298 |
and a reliable memory implement the unreliable memory. |
|
1299 |
*) |
|
1300 |
||
1301 |
(* Implementation of internal specification by combination of implementation |
|
1302 |
and history variable with explicit refinement mapping |
|
1303 |
*) |
|
1304 |
lemma Impl_IUSpec: "|- Implementation & Hist rmhist --> IUSpec memCh mm (resbar rmhist)" |
|
1305 |
by (auto simp: IUSpec_def Implementation_def IPImp_def MClkISpec_def |
|
1306 |
RPCISpec_def IRSpec_def Hist_def intro!: Step1 [temp_use] Step2 [temp_use]) |
|
1307 |
||
1308 |
(* The main theorem: introduce hiding and eliminate history variable. *) |
|
1309 |
lemma Implementation: "|- Implementation --> USpec memCh" |
|
1310 |
apply clarsimp |
|
1311 |
apply (frule History [temp_use]) |
|
1312 |
apply (auto simp: USpec_def intro: eexI [temp_use] Impl_IUSpec [temp_use] |
|
1313 |
MI_base [temp_use] elim!: eexE) |
|
1314 |
done |
|
3807 | 1315 |
|
1316 |
end |