public LemmaDecl GenerateBlockLemma(Block block, Block finalCfgBlock, IEnumerable <Block> finalCfgSuccessors,
string lemmaName, string vcHintsName)
{
var cmdsReduce = IsaBoogieTerm.RedCmdList(boogieContext,
IsaCommonTerms.TermIdentFromName(isaBlockInfo.CmdsQualifiedName(block)),
initState, finalState);
var vcAssm = vcinst.GetVCObjInstantiation(finalCfgBlock, declToVCMapping);
//do not use separate assumption, leads to issues
var conclusion = ConclusionBlock(finalCfgSuccessors, normalInitState, finalState, declToVCMapping, vcinst,
LemmaHelper.FinalStateIsMagic(block));
Term statement = TermBinary.MetaImplies(cmdsReduce, TermBinary.MetaImplies(vcAssm, conclusion));
var proof = BlockCorrectProof(block, finalCfgBlock, vcHintsName);
return(new LemmaDecl(lemmaName, ContextElem.CreateEmptyContext(), statement, proof));
}
private static HashSet <Block> ReachableBlocks(CFGRepr cfg)
{
var reachableBlocks = new HashSet <Block>();
var queue = new Queue <Block>();
reachableBlocks.Add(cfg.entry);
queue.Enqueue(cfg.entry);
while (queue.TryDequeue(out var b))
{
if (!LemmaHelper.FinalStateIsMagic(b))
{
//b's successors are not trivially unreachable
var successors = cfg.GetSuccessorBlocks(b);
foreach (var suc in successors)
{
reachableBlocks.Add(suc);
queue.Enqueue(suc);
}
}
}
return(reachableBlocks);
}
public LemmaDecl GenerateCfgLemma(
Block block,
Block finalCfgBlock,
bool isContainedInFinalCfg,
IEnumerable <Block> successors,
IEnumerable <Block> finalCfgSuccessors,
Term cfg,
Func <Block, string> cfgLemmaName,
LemmaDecl BlockLemma)
{
var red = IsaBoogieTerm.RedCFGMulti(
boogieContext,
cfg,
IsaBoogieTerm.CFGConfigNode(new NatConst(isaBlockInfo.BlockIds[block]),
IsaBoogieTerm.Normal(normalInitState)),
IsaBoogieTerm.CFGConfig(finalNode, finalState));
var assumption = new List <Term> {
red
};
var hasVcAssm = false;
if (isContainedInFinalCfg)
{
assumption.Add(vcinst.GetVCObjInstantiation(finalCfgBlock, declToVCMapping));
hasVcAssm = true;
}
else
{
//vc assumption is conjunction of reachable successors in final cfg
if (finalCfgSuccessors.Any())
{
assumption.Add(
LemmaHelper.ConjunctionOfSuccessorBlocks(finalCfgSuccessors, declToVCMapping, vcinst));
hasVcAssm = true;
}
}
Term conclusion = new TermBinary(finalState, IsaBoogieTerm.Failure(), TermBinary.BinaryOpCode.Neq);
var nodeLemma = isaBlockInfo.BlockCmdsMembershipLemma(block);
var outEdgesLemma = isaBlockInfo.OutEdgesMembershipLemma(block);
var proofMethods = new List <string>();
var eruleLocalBlock =
"erule " + (hasVcAssm ? ProofUtil.OF(BlockLemma.Name, "_", "assms(2)") : BlockLemma.Name);
if (isContainedInFinalCfg && LemmaHelper.FinalStateIsMagic(block))
{
proofMethods.Add("apply (rule converse_rtranclpE2[OF assms(1)], fastforce)");
proofMethods.Add(ProofUtil.Apply("rule " +
ProofUtil.OF("red_cfg_multi_backwards_step_magic", "assms(1)",
nodeLemma)));
proofMethods.Add(ProofUtil.By(eruleLocalBlock));
return(new LemmaDecl(cfgLemmaName(block), ContextElem.CreateWithAssumptions(assumption), conclusion,
new Proof(proofMethods)));
}
if (successors.Any())
{
proofMethods.Add("apply (rule converse_rtranclpE2[OF assms(1)], fastforce)");
var cfg_lemma = finalCfgSuccessors.Any()
? "red_cfg_multi_backwards_step"
: "red_cfg_multi_backwards_step_2";
proofMethods.Add(ProofUtil.Apply("rule " +
ProofUtil.OF(cfg_lemma, "assms(1)", nodeLemma)));
proofMethods.Add(ProofUtil.Apply(eruleLocalBlock));
proofMethods.Add("apply (" + ProofUtil.Simp(outEdgesLemma) + ")");
foreach (var bSuc in successors)
{
proofMethods.Add("apply (erule member_elim, simp)");
proofMethods.Add("apply (erule " + cfgLemmaName(bSuc) + ", simp?" + ")");
}
proofMethods.Add("by (simp add: member_rec(2))");
}
else
{
proofMethods.Add("apply (rule converse_rtranclpE2[OF assms(1)], fastforce)");
proofMethods.Add("apply (rule " + ProofUtil.OF("red_cfg_multi_backwards_step_no_succ", "assms(1)",
nodeLemma, outEdgesLemma) + ")");
if (isContainedInFinalCfg)
{
proofMethods.Add("using " + ProofUtil.OF(BlockLemma.Name, "_", "assms(2)") + " by blast");
}
else
{
proofMethods.Add("using " + BlockLemma.Name + " by blast");
}
}
return(new LemmaDecl(cfgLemmaName(block), ContextElem.CreateWithAssumptions(assumption), conclusion,
new Proof(proofMethods)));
}
