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pndfs_earlycycledetection.pvl
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/* * Optimised Parallel Nested DFS (with early cycle detection) */
class Worker {
/** ** Thread-local state */
Master master;
int tid; // thread identifier
int nthreads; // total number of threads (in `master`)
seq<int> color; // color set
boolean term; // thread-local termination flag
seq<boolean> observedred; // all states observed red by this worker
seq<boolean> pink; // set of pink states
boolean exploringred; // true if this worker is performing `dfsred`
boolean waiting; // if `exploringred == true`, then `waiting` indicates that the worker is done exploring and waiting for other threads to complete their exploration
int redroot; // if `exploringred == true`, then `redroot` is the root node from which a red DFS is being performed
// inline predicate that helps to connect the state of the master with the state of a worker.
static inline resource MasterState(Worker me, int N, seq<seq<boolean>> G, seq<boolean> A, int blueroot) =
me != null **
Perm(me.master, read) ** me.master != null **
Perm(me.nthreads, read) ** 0 < me.nthreads **
Perm(me.tid, read) ** 0 <= me.tid ** me.tid < me.nthreads **
Perm(me.term, 1/2) **
Perm(me.master.N, read) ** N == me.master.N **
Perm(me.master.G, read) ** G == me.master.G **
Perm(me.master.A, read) ** A == me.master.A **
Perm(me.master.blueroot, read) ** blueroot == me.master.blueroot **
Perm(me.master.threads, read) **
|me.master.threads| == me.nthreads **
me.master.threads[me.tid] == me;
/* ** Auxiliary operations */
context MasterState(this, N, G, A, blueroot);
context Master.AdjacencyMatrix(N, G) && |A| == N;
context Perm(observedred, 1/2) ** |observedred| == N;
context 0 <= s && s < N;
ensures (\forall int v; 0 <= v && v < N && v != s; observedred[v] == \old(observedred[v]));
ensures \old(observedred[s]) ==> observedred[s];
ensures term == \old(term);
void update_redmarkers(int N, seq<seq<boolean>> G, seq<boolean> A, int blueroot, int s) {
lock master;
if (master.red[s] && !observedred[s]) {
observedred = Master.UpdateBool(observedred, s, true);
}
unlock master;
}
//void update_redmarkers(int N, seq<seq<boolean>> G, seq<boolean> A, int blueroot, int s);
context MasterState(this, N, G, A, blueroot);
context Master.AdjacencyMatrix(N, G) && |A| == N;
context Perm(color, 1/2) ** |color| == N;
context Perm(pink, 1/2) ** |pink| == N;
context Perm(observedred, 1/2) ** |observedred| == N;
context 0 <= s && s < N;
context (\forall int v; 0 <= v && v < N && G[s][v]; color[v] == Master.blue() || color[v] == Master.cyan() || observedred[v]);
context color[s] == Master.cyan() || color[s] == Master.blue();
context A[s] ==> color[s] == Master.cyan();
context Perm(exploringred, 1/2) ** exploringred;
context Perm(waiting, 1/2) ** !waiting;
context Perm(redroot, 1/2);
context 0 <= redroot && redroot < N && A[redroot];
context A[s] ==> s == redroot;
context Master.ExPath(N, G, redroot, s, 1);
ensures (\forall int v; 0 <= v && v < N && v != s; pink[v] == \old(pink[v]));
ensures pink[s];
ensures color == \old(color);
ensures observedred == \old(observedred);
ensures redroot == \old(redroot);
ensures term == \old(term);
void update_setpink(int N, seq<seq<boolean>> G, seq<boolean> A, int blueroot, int s) {
lock master;
pink = Master.UpdateBool(pink, s, true);
unlock master;
}
//void update_setpink(int N, seq<seq<boolean>> G, seq<boolean> A, int blueroot, int s);
context MasterState(this, N, G, A, blueroot);
context Master.AdjacencyMatrix(N, G) && |A| == N;
context Perm(color, 1/2) ** |color| == N;
context Perm(pink, 1/2) ** |pink| == N;
context Perm(observedred, 1/2) ** |observedred| == N;
context 0 <= s && s < N && !A[s];
context color[s] == Master.blue();
context (\forall int v; 0 <= v && v < N && G[s][v]; (observedred[v] || pink[v]) && color[v] != Master.cyan()); // needed for invariant 2
requires pink[s];
ensures color == \old(color);
ensures term == \old(term);
ensures (\forall int v; 0 <= v && v < N && v != s; pink[v] == \old(pink[v]) && observedred[v] == \old(observedred[v]));
ensures observedred[s] && !pink[s];
void update_setred(int N, seq<seq<boolean>> G, seq<boolean> A, int blueroot, int s) {
lock master;
// if there exists a special path which we break by coloring `redroot` red, apply the lemma that finds another special path
assert (\forall int t1; 0 <= t1 && t1 < |master.threads| && Master.ExSpecialPath(N, G, master.threads[t1].color, master.threads[t1].pink, master.red);
Master.lemma_findspec_upd_thr(N, G, master.A, master.threads, master.red, Master.UpdateInt(master.redowner, s, tid), t1, s, master.threads[t1].redroot) &&
(\exists int t2; 0 <= t2 && t2 < |master.threads|; Master.ExSpecialPath(N, G, master.threads[t2].color, Master.UpdateBool(master.threads[t2].pink, s, false), Master.UpdateBool(master.red, s, true)))
);
master.red = Master.UpdateBool(master.red, s, true);
master.redowner = Master.UpdateInt(master.redowner, s, tid); // ghostcode
observedred = Master.UpdateBool(observedred, s, true); // ghostcode
pink = Master.UpdateBool(pink, s, false);
// increases performance (does not make much difference yet)
assert Master.KeyInvariant(N, G, A, master.threads, master.red, blueroot);
unlock master;
}
//void update_setred(int N, seq<seq<boolean>> G, seq<boolean> A, int blueroot, int s);
invariant MasterState(this, N, G, A, blueroot);
invariant Master.AdjacencyMatrix(N, G) && |A| == N;
invariant Perm(color, 1/2) ** |color| == N;
invariant Perm(pink, 1/2) ** |pink| == N;
invariant Perm(observedred, 1/2) ** |observedred| == N;
invariant Perm(exploringred, 1/2) ** Perm(waiting, 1/2) ** Perm(redroot, 1/2);
invariant 0 <= redroot && redroot < N;
invariant (\forall int v; 0 <= v && v < N && G[redroot][v]; observedred[v] && color[v] != Master.cyan()); // needed for invariant 2
requires exploringred && waiting;
ensures color == \old(color);
ensures redroot == \old(redroot);
ensures term == \old(term);
ensures (\forall int v; 0 <= v && v < N && v != redroot; pink[v] == \old(pink[v]) && observedred[v] == \old(observedred[v]));
ensures observedred[redroot] && !pink[redroot];
ensures !waiting && !exploringred;
void update_setred_await(int N, seq<seq<boolean>> G, seq<boolean> A, int blueroot) {
boolean stop = false;
loop_invariant color == \old(color);
loop_invariant redroot == \old(redroot);
loop_invariant term == \old(term);
loop_invariant !stop ==> (exploringred && waiting);
loop_invariant !stop ==> pink == \old(pink);
loop_invariant !stop ==> observedred == \old(observedred);
loop_invariant stop ==> (\forall int v; 0 <= v && v < N && v != redroot; pink[v] == \old(pink[v]));
loop_invariant stop ==> (\forall int v; 0 <= v && v < N && v != redroot; observedred[v] == \old(observedred[v]));
loop_invariant stop ==> (observedred[redroot] && !pink[redroot]);
loop_invariant stop ==> (!waiting && !exploringred);
while (!stop) {
lock master;
// the following if-statement can also be implemented using another while-loop if one does not use PVL to implement the algorithm, but for now this is more convenient
if ((\forall int t; 0 <= t && t < |master.threads| && master.threads[t].exploringred && master.threads[t].redroot == redroot; master.threads[t].waiting)) {
// if `redroot` is not already red but there exists a cycle that includes `redroot`, we can also find a special path using our lemma
assert (!master.red[redroot] && Master.ExPath(N, G, redroot, redroot, 2)) ==> (
Master.lemma_findspec_cycle_thr(N, G, master.A, master.threads, master.red, master.redowner, redroot) &&
(\exists int t; 0 <= t && t < |master.threads|; Master.ExSpecialPath(N, G, master.threads[t].color, master.threads[t].pink, master.red))
);
// if there exists a special path which we break by coloring `redroot` red, apply the lemma that finds another special path
assert (\forall int t1; 0 <= t1 && t1 < |master.threads| && Master.ExSpecialPath(N, G, master.threads[t1].color, master.threads[t1].pink, master.red);
Master.lemma_findspec_upd_thr(N, G, master.A, master.threads, master.red, Master.UpdateInt(master.redowner, redroot, tid), t1, redroot, master.threads[t1].redroot) &&
(\exists int t2; 0 <= t2 && t2 < |master.threads|; Master.ExSpecialPath(N, G, master.threads[t2].color, Master.UpdateBool(master.threads[t2].pink, redroot, false), Master.UpdateBool(master.red, redroot, true)))
);
master.red = Master.UpdateBool(master.red, redroot, true);
master.redowner = Master.UpdateInt(master.redowner, redroot, tid); // ghost
observedred = Master.UpdateBool(observedred, redroot, true); // ghost
pink = Master.UpdateBool(pink, redroot, false);
exploringred = false; // ghost
waiting = false; // ghost
stop = true;
// increases verification performance
assert (\forall int t; 0 <= t && t < |master.threads|;
master.ColorTypes(master.threads[t].color) &&
master.ThreadLocalInvariants(N, G, A, master.threads[t].color, master.threads[t].pink, master.red)
);
// invariant 2; increases performance
assert (\forall int v; 0 <= v && v < N && master.red[v];
(\forall int w; 0 <= w && w < N && G[v][w];
master.red[w] || (master.threads[master.redowner[v]].pink[w] && master.threads[master.redowner[v]].color[w] != Master.cyan())
)
);
// increases verification performance
assert Master.Invariant6(N, G, A, master.red);
assert Master.KeyInvariant(N, G, A, master.threads, master.red, blueroot);
}
unlock master;
}
}
//void update_setred_await(int N, seq<seq<boolean>> G, seq<boolean> A, int blueroot);
context MasterState(this, N, G, A, blueroot);
context Master.AdjacencyMatrix(N, G) ** |A| == N;
context Perm(pink, 1/2) ** |pink| == N;
context Perm(color, 1/2) ** |color| == N;
context Perm(exploringred, 1/2) ** !exploringred;
context 0 <= s && s < N;
context Master.ColorTypes(color);
requires !pink[s];
ensures pink == \old(pink);
ensures term == \old(term);
ensures (\forall int v; 0 <= v && v < N && v != s; color[v] == \old(color[v]));
ensures color[s] == Master.cyan();
void update_setcyan(int N, seq<seq<boolean>> G, seq<boolean> A, int blueroot, int s) {
lock master;
color = Master.UpdateInt(color, s, Master.cyan());
unlock master;
}
//void update_setcyan(int N, seq<seq<boolean>> G, seq<boolean> A, int blueroot, int s);
context MasterState(this, N, G, A, blueroot);
context Master.AdjacencyMatrix(N, G) ** |A| == N;
context Perm(color, 1/2) ** |color| == N;
context Perm(pink, 1/2) ** |pink| == N;
context Perm(observedred, 1/2) ** |observedred| == N;
context 0 <= s && s < N;
context Master.ColorTypes(color);
context (\forall int v; 0 <= v && v < N && G[s][v]; color[v] == Master.blue() || color[v] == Master.cyan() || observedred[v]);
context (\forall int v; 0 <= v && v < N; !pink[v]);
context A[s] ==> observedred[s];
context !pink[s];
requires color[s] == Master.cyan();
ensures (\forall int v; 0 <= v && v < N && v != s; color[v] == \old(color[v]));
ensures color[s] == Master.blue();
ensures pink == \old(pink);
ensures observedred == \old(observedred);
ensures term == \old(term);
void update_setblue(int N, seq<seq<boolean>> G, seq<boolean> A, int blueroot, int s) {
lock master;
color = Master.UpdateInt(color, s, Master.blue());
unlock master;
}
//void update_setblue(int N, seq<seq<boolean>> G, seq<boolean> A, int blueroot, int s);
context MasterState(this, N, G, A, blueroot);
context Master.AdjacencyMatrix(N, G) && |A| == N;
context Perm(pink, 1/2) ** |pink| == N;
context Perm(color, 1/2) ** |color| == N;
context Perm(observedred, 1/2) ** |observedred| == N;
context Perm(waiting, 1/2) ** Perm(redroot, 1/2);
context 0 <= redroot && redroot < N;
context pink[redroot] && A[redroot];
context (\forall int v; 0 <= v && v < N && v != redroot; !pink[v]);
context (\forall int v; 0 <= v && v < N && G[redroot][v]; observedred[v]);
context (\forall int v; 0 <= v && v < N && G[redroot][v]; color[v] != Master.cyan());
requires !waiting;
ensures waiting;
ensures redroot == \old(redroot);
ensures color == \old(color);
ensures observedred == \old(observedred);
ensures pink == \old(pink);
ensures redroot == \old(redroot);
ensures term == \old(term);
void update_setwaiting(int N, seq<seq<boolean>> G, seq<boolean> A, int blueroot) {
lock master;
waiting = true;
unlock master;
}
//void update_setwaiting(int N, seq<seq<boolean>> G, seq<boolean> A, int blueroot);
context MasterState(this, N, G, A, blueroot);
context Master.AdjacencyMatrix(N, G) && |A| == N;
context Perm(color, 1/2) ** |color| == N;
context Perm(observedred, 1/2) ** |observedred| == N;
context Perm(exploringred, 1/2) ** Perm(waiting, 1/2) ** Perm(redroot, 1/2);
context 0 <= root && root < N && A[root];
context (\forall int v; 0 <= v && v < N && color[v] == Master.cyan(); Master.ExPath(N, G, v, root, 1));
requires !exploringred && !observedred[root];
ensures color == \old(color);
ensures term == \old(term);
ensures (\forall int v; 0 <= v && v < N && v != root; observedred[v] == \old(observedred[v]));
ensures !observedred[root] ==> (exploringred && !waiting && redroot == root);
ensures observedred[root] ==> (!exploringred && redroot == \old(redroot));
void update_explorered(int N, seq<seq<boolean>> G, seq<boolean> A, int blueroot, int root) {
lock master;
if (master.red[root]) {
observedred = Master.UpdateBool(observedred, root, true);
}
else {
exploringred = true;
waiting = false;
redroot = root;
}
unlock master;
}
//void update_explorered(int N, seq<seq<boolean>> G, seq<boolean> A, int blueroot, int root);
context MasterState(this, N, G, A, blueroot);
ensures !\result ==> (term == \old(term));
ensures \result ==> term;
boolean update_abort(int N, seq<seq<boolean>> G, seq<boolean> A, int blueroot) {
boolean aborting = false;
lock master;
if (master.abort) {
term = true;
aborting = true;
}
unlock master;
return aborting;
}
//boolean update_abort(int N, seq<seq<boolean>> G, seq<boolean> A, int blueroot);
/* ** DFS red */
yields int c; // explicit witness
yields seq<int> P; // explicit witness
invariant MasterState(this, N, G, A, blueroot);
invariant Master.AdjacencyMatrix(N, G) && |A| == N;
invariant Perm(color, 1/2) ** |color| == N;
invariant Perm(pink, 1/2) ** |pink| == N;
invariant Perm(observedred, 1/2) ** |observedred| == N;
invariant 0 <= s && s < N;
invariant 0 <= root && root < N;
invariant Master.ColorTypes(color);
invariant s == root || color[s] == Master.blue();
invariant color[root] == Master.cyan() && A[root];
invariant Master.Invariant1a(N, G, color, observedred);
invariant Master.Invariant1b(N, G, color, pink, observedred);
invariant (\forall int v; 0 <= v && v < N && G[root][v]; color[v] == Master.blue() || color[v] == Master.cyan() || observedred[v]); // `Next({root}, Blue ∪ Cyan ∪ Red)`
invariant (\forall int v; 0 <= v && v < N; !pink[v] || !observedred[v]); // `Pink ∩ Red = Ø`
invariant Master.Invariant3(N, G, A, color, observedred);
invariant Master.Invariant4(N, G, A, color, pink);
invariant (\forall int v; 0 <= v && v < N && pink[v] && v != root; color[v] == Master.blue()); // `Pink ⊆ Blue ∪ {root}` -- L5 (improved)
invariant (\forall int v; 0 <= v && v < N && v != root && observedred[v]; color[v] != Master.cyan()); // `Red - {root} ⊆ (G - {root}) - Cyan`
invariant (\forall int v; 0 <= v && v < N && color[v] == Master.cyan(); Master.ExPath(N, G, v, root, 1));
invariant (\forall int v; 0 <= v && v < N && pink[v]; Master.ExPath(N, G, root, v, 1));
invariant A[s] ==> s == root;
invariant s == root ==> A[s];
invariant Perm(exploringred, 1/2);
invariant Perm(waiting, 1/2) ** !waiting;
invariant Perm(redroot, 1/2) ** redroot == root;
requires !pink[s];
requires !observedred[s];
requires !term;
requires s == root ==> (\forall int v; 0 <= v && v < N && v != root; !pink[v]);
requires exploringred;
requires Master.ExPath(N, G, root, s, 1);
ensures color == \old(color);
ensures (\forall int v; 0 <= v && v < N && \old(observedred[v]); observedred[v]); // `old(Red) ⊆ Red`
ensures \result == Master.found() || \result == Master.notfound() || \result == Master.aborted(); // return types
ensures \result == Master.notfound() ==> (\forall int v; 0 <= v && v < N; pink[v] == \old(pink[v])); // `¬found ⇒ old(Pink) = Pink`
ensures \result == Master.notfound() ==> observedred[s];
ensures (\result == Master.notfound() && s == root) ==> (\forall int v; 0 <= v && v < N && v != root; !pink[v]);
ensures \result == Master.notfound() ==> (A[s] == !exploringred);
ensures \result == Master.found() ==> (0 <= c && c < N && color[c] == Master.cyan());
ensures \result == Master.found() ==> (1 < |P| && Master.Path(N, G, s, c, P));
ensures \result == Master.aborted() ==> term;
ensures \result != Master.aborted() ==> !term;
int dfsred(int N, seq<seq<boolean>> G, seq<boolean> A, int blueroot, int s, int root) {
update_setpink(N, G, A, blueroot, s);
assert A[s] ==> s == root;
int found = Master.notfound();
int t = 0;
assert !term;
// The parallel algorithm should actually traverse the successors of `s` in random order to obtain any speedup.
// However, from a verification point-of-view this does not matter, as long as all successors have been
// considered when the while-loop below terminates.
loop_invariant 0 <= t && t <= N;
loop_invariant color == \old(color);
loop_invariant (\forall int v; 0 <= v && v < N && \old(observedred[v]); observedred[v]); // `old(Red) ⊆ Red`
loop_invariant found == Master.found() || found == Master.notfound() || found == Master.aborted(); // return types
loop_invariant found == Master.notfound() ==> pink[s]; // `s ∈ Pink`
loop_invariant found == Master.notfound() ==> (\forall int v; 0 <= v && v < N && v != s; pink[v] == \old(pink[v])); // `¬found ⇒ old(Pink - {s}) = Pink - {s}`
loop_invariant found == Master.notfound() ==> (\forall int v; 0 <= v && v < t && G[s][v]; observedred[v] || pink[v]); // needed for invariant 2
loop_invariant found == Master.notfound() ==> (\forall int v; 0 <= v && v < t && G[s][v]; color[v] != Master.cyan()); // needed for invariant 2
loop_invariant (found == Master.notfound() && s == root) ==> (\forall int v; 0 <= v && v < N && v != root; !pink[v]);
loop_invariant found == Master.notfound() ==> exploringred;
loop_invariant found == Master.found() ==> (0 <= c && c < N && color[c] == Master.cyan());
loop_invariant found == Master.found() ==> (1 < |P| && Master.Path(N, G, s, c, P));
loop_invariant found == Master.aborted() ==> term;
loop_invariant found != Master.aborted() ==> !term;
while (t < N && found == Master.notfound()) {
// check for global termination
boolean aborting = update_abort(N, G, A, blueroot);
if (aborting) {
found = Master.aborted();
}
if (G[s][t] && found != Master.aborted()) {
if (color[t] == Master.cyan()) {
c = t;
P = seq<int> { s, t };
found = Master.found();
}
else {
if (!pink[t]) {
update_redmarkers(N, G, A, blueroot, t);
if (!observedred[t]) {
assert !A[t];
Master.lemma_edge_path_m(N, G, s, t);
Master.lemma_expath_trans_m(N, G, root, s, t, 1, 1);
found = dfsred(N, G, A, blueroot, t, root) with { c = c; P = P; };
if (found == Master.found()) {
P = seq<int> { s } + P;
}
}
}
}
}
t = t + 1;
}
if (found == Master.notfound()) {
if (A[s]) {
assert s == root;
update_setwaiting(N, G, A, blueroot);
update_setred_await(N, G, A, blueroot);
}
else {
update_setred(N, G, A, blueroot, s);
}
}
return found;
}
//int dfsred(int N, seq<seq<boolean>> G, seq<boolean> A, int blueroot, int s, int root);
/* ** DFS blue */
/// verification takes: 56997 ms
/// verification takes: 58683 ms (with termination handling)
yields int a; // explicit witness
yields seq<int> P; // explicit witness
invariant MasterState(this, N, G, A, blueroot);
invariant Master.AdjacencyMatrix(N, G) && |A| == N;
invariant Perm(color, 1/2) ** |color| == N;
invariant Perm(pink, 1/2) ** |pink| == N;
invariant Perm(observedred, 1/2) ** |observedred| == N;
invariant Perm(exploringred, 1/2);
invariant Perm(waiting, 1/2);
invariant Perm(redroot, 1/2);
invariant 0 <= s && s < N;
invariant 0 <= blueroot && blueroot < N;
invariant Master.ColorTypes(color);
invariant Master.Invariant1b(N, G, color, pink, observedred);
invariant Master.Invariant3(N, G, A, color, observedred);
requires color[s] == Master.white();
requires !observedred[s];
requires (\forall int v; 0 <= v && v < N; !pink[v]); // `Pink = Ø`
requires Master.Invariant1a(N, G, color, observedred);
requires (\forall int v; 0 <= v && v < N && observedred[v]; color[v] != Master.cyan()); // `Red ⊆ G - Cyan`
requires !exploringred;
requires !term;
requires (\forall int v; 0 <= v && v < N && color[v] == Master.cyan(); Master.ExPath(N, G, v, s, 1));
ensures (\forall int v; 0 <= v && v < N && \old(color[v]) == Master.blue(); color[v] == Master.blue()); // `old(Blue) ⊆ Blue`
ensures (\forall int v; 0 <= v && v < N && \old(observedred[v]); observedred[v]); // `old(Red) ⊆ Red`
ensures \result == Master.found() || \result == Master.notfound() || \result == Master.aborted(); // return types
ensures \result == Master.notfound() ==> (\forall int v; 0 <= v && v < N && color[v] == Master.cyan(); \old(color[v]) == Master.cyan());
ensures \result == Master.notfound() ==> (\forall int v; 0 <= v && v < N && \old(color[v]) == Master.cyan(); color[v] == Master.cyan());
ensures \result == Master.notfound() ==> color[s] == Master.blue();
ensures \result == Master.notfound() ==> (\forall int v; 0 <= v && v < N; !pink[v]); // `Pink = Ø`
ensures \result == Master.notfound() ==> Master.Invariant1a(N, G, color, observedred);
ensures \result == Master.notfound() ==> (\forall int v; 0 <= v && v < N && observedred[v]; color[v] != Master.cyan()); // `Red ⊆ G - Cyan`
ensures \result == Master.notfound() ==> !exploringred;
ensures \result == Master.notfound() ==> (\forall int v; 0 <= v && v < N && color[v] == Master.cyan(); Master.ExPath(N, G, v, s, 1));
ensures \result == Master.found() ==> (0 <= a && a < N && A[a]);
ensures \result == Master.found() ==> (Master.Path(N, G, s, a, P) && Master.ExPath(N, G, a, a, 2));
ensures \result == Master.aborted() ==> term;
ensures \result != Master.aborted() ==> !term;
int dfsblue(int N, seq<seq<boolean>> G, seq<boolean> A, int blueroot, int s) {
update_setcyan(N, G, A, blueroot, s);
assert Master.ExPath(N, G, s, s, |seq<int> { s }|);
int found = Master.notfound();
int t = 0;
// The parallel algorithm should actually traverse the successors of `s` in random order to obtain any speedup.
// However, from a verification point-of-view this is irrelevant, as long as all successors have been
// considered when the while-loop below terminates.
loop_invariant 0 <= t && t <= N;
loop_invariant (\forall int v; 0 <= v && v < N && \old(color[v]) == Master.blue(); color[v] == Master.blue()); // `old(Blue) ⊆ Blue`
loop_invariant (\forall int v; 0 <= v && v < N && \old(observedred[v]); observedred[v]); // `old(Red) ⊆ Red`
loop_invariant found == Master.found() || found == Master.notfound() || found == Master.aborted(); // return types
loop_invariant found == Master.notfound() ==> (\forall int i; 0 <= i && i < N && i != s; (color[i] == Master.cyan()) == (\old(color[i]) == Master.cyan()));
loop_invariant found == Master.notfound() ==> color[s] == Master.cyan();
loop_invariant found == Master.notfound() ==> (\forall int v; 0 <= v && v < t && G[s][v]; color[v] == Master.cyan() || color[v] == Master.blue() || observedred[v]);
loop_invariant found == Master.notfound() ==> (\forall int v; 0 <= v && v < N; !pink[v]);
loop_invariant found == Master.notfound() ==> Master.Invariant1a(N, G, color, observedred);
loop_invariant found == Master.notfound() ==> (\forall int v; 0 <= v && v < N && observedred[v]; color[v] != Master.cyan()); // `¬found ⇒ Red ⊆ G - Cyan`
loop_invariant found == Master.notfound() ==> !exploringred;
loop_invariant found == Master.notfound() ==> (\forall int v; 0 <= v && v < N && color[v] == Master.cyan(); Master.ExPath(N, G, v, s, 1));
loop_invariant found == Master.found() ==> (0 <= a && a < N && A[a]);
loop_invariant found == Master.found() ==> (Master.Path(N, G, s, a, P) && Master.ExPath(N, G, a, a, 2));
loop_invariant found == Master.aborted() ==> term;
loop_invariant found != Master.aborted() ==> !term;
while (t < N && found == Master.notfound()) {
// check for global termination
boolean aborting = update_abort(N, G, A, blueroot);
if (aborting) {
found = Master.aborted();
}
if (G[s][t] && found != Master.aborted()) {
if (color[t] == Master.cyan() && (A[s] || A[t])) {
found = Master.found(); // early cycle detection
if (A[s]) {
a = s;
P = seq<int> { a };
Master.lemma_edge_path_m(N, G, a, t);
Master.lemma_expath_trans_m(N, G, a, t, a, 2, 1);
}
else {
assert A[t];
a = t;
P = seq<int> { s, a };
Master.lemma_edge_path_m(N, G, s, a);
Master.lemma_expath_trans_m(N, G, a, s, a, 2, 1);
}
}
if (found == Master.notfound()) {
if (color[t] == Master.white()) {
update_redmarkers(N, G, A, blueroot, t);
if (!observedred[t]) {
Master.lemma_nextcyan_m(N, G, color, s, t);
found = dfsblue(N, G, A, blueroot, t) with { a = a; P = P; };
if (found == Master.found()) {
P = seq<int> { s } + P;
}
}
}
}
}
t = t + 1;
}
if (found == Master.notfound() && A[s]) {
update_explorered(N, G, A, blueroot, s);
assert (\forall int v; 0 <= v && v < N && v != s && observedred[v]; color[v] != Master.cyan()); // needed
if (!observedred[s]) {
int c;
seq<int> Q;
found = dfsred(N, G, A, blueroot, s, s) with { c = c; P = Q; };
if (found == Master.found()) {
Master.lemma_cyclefoundhere_m(N, G, color, s, c, Q);
a = s;
P = seq<int> { s };
}
}
}
if (found == Master.notfound()) {
update_setblue(N, G, A, blueroot, s);
}
return found;
}
//int dfsblue(int N, seq<seq<boolean>> G, seq<boolean> A, int blueroot, int s);
}
class Master {
/** ** Global state */
int N; // total number of states
seq<seq<boolean>> G; // adjacency matrix representation of the input graph
seq<boolean> A; // the (finite) set of accepting states
seq<boolean> red; // set of red states
seq<int> redowner; // for bookkeeping: if `red[v] == true`, then `redowner[v]` contains the rank of the worker that has marked node `v` red
seq<Worker> threads; // worker set
boolean abort; // global termination flag
int blueroot; // the initial state from which all threads perform NDFS
int result; // result of NDFS
resource lock_invariant() =
// graph state
Perm(N, read) ** Perm(G, read) ** AdjacencyMatrix(N, G) **
Perm(A, read) ** |A| == N **
Perm(blueroot, read) ** 0 <= blueroot ** blueroot < N **
Perm(red, 1) ** |red| == N **
Perm(redowner, 1) ** |redowner| == N **
// state of threads
Perm(threads, read) ** 0 < |threads| **
(\forall int t; 0 <= t && t < |threads|;
threads[t] != null
) **
(\forall* int t; 0 <= t && t < |threads|;
Perm(threads[t].master, read) **
Perm(threads[t].tid, read) **
Perm(threads[t].nthreads, read) **
Perm(threads[t].color, 1/2) **
Perm(threads[t].observedred, 1/2) **
Perm(threads[t].pink, 1/2) **
Perm(threads[t].exploringred, 1/2) **
Perm(threads[t].waiting, 1/2) **
Perm(threads[t].term, 1/2) **
Perm(threads[t].redroot, 1/2)
) **
(\forall int t; 0 <= t && t < |threads|;
threads[t].master == this &&
threads[t].tid == t &&
threads[t].nthreads == |threads|
) **
(\forall int t; 0 <= t && t < |threads|;
|threads[t].color| == N &&
|threads[t].observedred| == N &&
|threads[t].pink| == N
) **
// state of red nodes
(\forall int t; 0 <= t && t < |threads|;
(\forall int v; 0 <= v && v < N && threads[t].observedred[v]; red[v])
) **
(\forall int v; 0 <= v && v < N && red[v];
0 <= redowner[v] && redowner[v] < |threads| &&
threads[redowner[v]].observedred[v] &&
threads[redowner[v]].color[v] != white()
) **
// PNDFS invariants
(\forall int t; 0 <= t && t < |threads|;
ColorTypes(threads[t].color) &&
ThreadLocalInvariants(N, G, A, threads[t].color, threads[t].pink, red)
) **
(\forall int v; 0 <= v && v < N && red[v];
(\forall int w; 0 <= w && w < N && G[v][w];
red[w] || (threads[redowner[v]].pink[w] && threads[redowner[v]].color[w] != cyan()) // Invariant 2
)
) **
Invariant6(N, G, A, red) **
KeyInvariant(N, G, A, threads, red, blueroot) **
// `dfsred` worker statusses
(\forall int t; 0 <= t && t < |threads| && threads[t].exploringred; 0 <= threads[t].redroot && threads[t].redroot < N) **
(\forall int t; 0 <= t && t < |threads| && threads[t].exploringred; A[threads[t].redroot]) **
(\forall int t; 0 <= t && t < |threads| && threads[t].exploringred && !threads[t].waiting; !red[threads[t].redroot]) **
(\forall int t; 0 <= t && t < |threads| && threads[t].exploringred && threads[t].waiting; threads[t].pink[threads[t].redroot]) **
(\forall int t; 0 <= t && t < |threads| && threads[t].exploringred && threads[t].waiting; (\forall int v; 0 <= v && v < N && v != threads[t].redroot; !threads[t].pink[v])) **
(\forall int t; 0 <= t && t < |threads| && !threads[t].exploringred; (\forall int v; 0 <= v && v < N; !threads[t].pink[v])) **
(\forall int t; 0 <= t && t < |threads|; (\forall int p; 0 <= p && p < N && threads[t].pink[p] && A[p]; threads[t].exploringred && threads[t].redroot == p)) **
(\forall int t; 0 <= t && t < |threads| && threads[t].exploringred; (\forall int p; 0 <= p && p < N && threads[t].pink[p]; Master.ExPath(N, G, threads[t].redroot, p, 1))) **
(\forall int t; 0 <= t && t < |threads| && threads[t].exploringred; (\forall int c; 0 <= c && c < N && threads[t].color[c] == Master.cyan(); Master.ExPath(N, G, c, threads[t].redroot, 1))) **
(\forall int t; 0 <= t && t < |threads| && threads[t].exploringred && threads[t].waiting; (\forall int v; 0 <= v && v < N && G[threads[t].redroot][v]; red[v] && threads[t].color[v] != Master.cyan())) **
// termination handling
Perm(abort, 1/2) **
(\forall int t; 0 <= t && t < |threads|; threads[t].term ==> abort);
requires 0 < nthreads;
requires AdjacencyMatrix(N_, G_) && |A_| == N_;
requires 0 <= root && root < N_;
ensures Perm(N, read) ** N == N_;
ensures Perm(G, read) ** G == G_;
ensures Perm(A, read) ** A == A_;
ensures AdjacencyMatrix(N, G) && |A| == N;
ensures Perm(blueroot, read) ** blueroot == root;
ensures 0 <= blueroot && blueroot < N;
ensures Perm(threads, read) ** |threads| == nthreads;
ensures Perm(abort, 1/2) ** !abort;
ensures (\forall* int t; 0 <= t && t < nthreads;
Perm(threads[t].master, read) ** threads[t].master == this **
Perm(threads[t].tid, read) ** threads[t].tid == t **
Perm(threads[t].nthreads, read) ** threads[t].nthreads == nthreads **
Perm(threads[t].color, 1/2) ** threads[t].color == NewIntSeq(N) **
Perm(threads[t].observedred, 1/2) ** threads[t].observedred == NewBoolSeq(N) **
Perm(threads[t].pink, 1/2) ** threads[t].pink == NewBoolSeq(N) **
Perm(threads[t].exploringred, 1/2) ** !threads[t].exploringred **
Perm(threads[t].waiting, 1/2) ** !threads[t].waiting **
Perm(threads[t].term, 1/2) ** !threads[t].term **
Perm(threads[t].redroot, 1/2)
);
Master(int N_, seq<seq<boolean>> G_, seq<boolean> A_, int root, int nthreads) {
N = N_;
G = G_;
A = A_;
blueroot = root;
red = NewBoolSeq(N);
redowner = NewIntSeq(N);
threads = seq<Worker> { };
int tid = 0;
// construct the worker set
loop_invariant 0 <= tid && tid <= nthreads;
loop_invariant Perm(N, 1) ** N == N_;
loop_invariant Perm(G, 1) ** G == G_;
loop_invariant Perm(A, 1) ** A == A_;
loop_invariant Perm(red, 1) ** red == NewBoolSeq(N);
loop_invariant Perm(redowner, 1) ** redowner == NewIntSeq(N);
loop_invariant AdjacencyMatrix(N, G) && |A| == N;
loop_invariant 0 <= root && root < N;
loop_invariant Perm(blueroot, 1) ** blueroot == root;
loop_invariant Perm(threads, 1) ** |threads| == tid;
loop_invariant (\forall* int t; 0 <= t && t < tid;
Perm(threads[t].master, 1) ** threads[t].master == this **
Perm(threads[t].tid, 1) ** threads[t].tid == t **
Perm(threads[t].nthreads, 1) ** threads[t].nthreads == nthreads **
Perm(threads[t].color, 1) ** threads[t].color == NewIntSeq(N) **
Perm(threads[t].observedred, 1) ** threads[t].observedred == NewBoolSeq(N) **
Perm(threads[t].pink, 1) ** threads[t].pink == NewBoolSeq(N) **
Perm(threads[t].exploringred, 1) ** !threads[t].exploringred **
Perm(threads[t].waiting, 1) ** !threads[t].waiting **
Perm(threads[t].term, 1) ** !threads[t].term **
Perm(threads[t].redroot, 1)
);
loop_invariant (\forall int t; 0 <= t && t < tid;
ColorTypes(threads[t].color) &&
ThreadLocalInvariants(N, G, A, threads[t].color, threads[t].pink, red)
);
while (tid < nthreads) {
Worker thread = new Worker();
thread.master = this;
thread.tid = tid;
thread.nthreads = nthreads;
thread.color = NewIntSeq(N);
thread.observedred = NewBoolSeq(N);
thread.pink = NewBoolSeq(N);
thread.exploringred = false;
thread.waiting = false;
thread.term = false;
threads = threads + seq<Worker> { thread };
tid = tid + 1;
}
abort = false;
}
/* ** NDFS color codes */
static pure int white() = 0;
static pure int cyan() = 1;
static pure int blue() = 2;
/* ** NDFS return types */
static pure int notfound() = 0;
static pure int found() = 1;
static pure int aborted() = 2;
/** ** Default predicates */
// determines whether `G` is an adjacency matrix of size `N`
static inline pure boolean AdjacencyMatrix(int N, seq<seq<boolean>> G) =
|G| == N && (\forall seq<boolean> e; e in G; |e| == N);
// determines whether the given color set has the proper colors
static inline pure boolean ColorTypes(seq<int> color) =
(\forall int i; 0 <= i && i < |color|; color[i] == white() || color[i] == cyan() || color[i] == blue());
// determines whether `P` is a path from `x` to `y` in the graph `G`.
requires AdjacencyMatrix(N, G);
static pure boolean Path(int N, seq<seq<boolean>> G, int x, int y, seq<int> P) =
0 <= x && x < N && 0 <= y && y < N &&
0 < |P| && P[0] == x && P[|P| - 1] == y &&
(\forall int j; 0 <= j && j < |P|; 0 <= P[j] && P[j] < N) &&
(\forall int j; 0 <= j && j < |P| - 1; G[P[j]][P[j + 1]]);
// Shorthand notation for paths.
static pure inline boolean Path(int N, seq<seq<boolean>> G, seq<int> P) = Path(N, G, P[0], P[|P|-1], P);
// existential quantification over paths in `G` of length at least `len`.
requires AdjacencyMatrix(N, G);
static pure boolean ExPath(int N, seq<seq<boolean>> G, int x, int y, int len) =
(\exists seq<int> P; len <= |P|; Path(N, G, x, y, P));
// A _special path_ is defined to be a path `P` starting with a pink node and ending with a cyan node, so that all states on `P` are not red.
requires AdjacencyMatrix(N, G) && |color| == N && |pink| == N && |red| == N;
requires 1 < |P| && Path(N, G, P);
static pure boolean SpecialPath(int N, seq<seq<boolean>> G, seq<int> P, seq<int> color, seq<boolean> pink, seq<boolean> red) =
pink[P[0]] && color[P[|P| - 1]] == cyan() && (\forall int i; 0 <= i && i < |P|; !red[P[i]]);
// Existential quantification of "special" paths
requires AdjacencyMatrix(N, G) && |color| == N && |pink| == N && |red| == N;
static pure boolean ExSpecialPath(int N, seq<seq<boolean>> G, seq<int> color, seq<boolean> pink, seq<boolean> red) =
(\exists seq<int> P; 1 < |P| && Path(N, G, P); SpecialPath(N, G, P, color, pink, red));
// Determines the existence of an accepting cycle from `root` to the accepting state `a`
requires AdjacencyMatrix(N, G) && |A| == N;
requires 0 <= root && root < N;
requires 0 <= a && a < N && A[a];
static pure boolean ExAccCycle(int N, seq<seq<boolean>> G, seq<boolean> A, int root, int a) =
ExPath(N, G, root, a, 1) && ExPath(N, G, a, a, 2);
/* ** Default operations */
// some workaround definitions
static inline pure boolean GetBool(seq<boolean> xs, int i) = xs[i];
static inline pure int GetInt(seq<int> xs, int i) = xs[i];
requires 0 <= i && i < |xs|;
ensures |\result| == |xs|;
ensures \result[i] == v;
ensures (\forall int j; 0 <= j && j < |xs| && j != i; \result[j] == xs[j]);
pure static seq<int> UpdateInt(seq<int> xs, int i, int v) =
0 < i ? seq<int> { head(xs) } + UpdateInt(tail(xs), i - 1, v) : seq<int> { v } + tail(xs);
requires 0 <= i && i < |xs|;
ensures |\result| == |xs|;
ensures \result[i] == v;
ensures (\forall int j; 0 <= j && j < |xs| && j != i; \result[j] == xs[j]);
pure static seq<boolean> UpdateBool(seq<boolean> xs, int i, boolean v) =
0 < i ? seq<boolean> { head(xs) } + UpdateBool(tail(xs), i - 1, v) : seq<boolean> { v } + tail(xs);
// constructs a fresh color set containing `n` white colors.
requires 0 <= n;
ensures |\result| == n;
ensures (\forall int i; 0 <= i && i < n; \result[i] == white());
pure static seq<int> NewIntSeq(int n) =
0 < n ? seq<int> { white() } + NewIntSeq(n - 1) : seq<int> { };
// constructs a fresh color set containing `n` white colors.
requires 0 <= n;
ensures |\result| == n;
ensures (\forall int i; 0 <= i && i < n; !\result[i]);
pure static seq<boolean> NewBoolSeq(int n) =
0 < n ? seq<boolean> { false } + NewBoolSeq(n - 1) : seq<boolean> { };
// skips the first `n` elements of the integer sequence `xs`.
requires 0 <= n && n <= |xs|;
ensures |\result| == |xs| - n;
ensures (\forall int i; 0 <= i && i < |xs| - n; xs[n + i] == \result[i]);
pure static seq<int> SkipInt(seq<int> xs, int n) =
0 < n ? SkipInt(tail(xs), n - 1) : xs;
// takes the first `n` elements of the integer sequence `xs`.
requires 0 <= n && n <= |xs|;
ensures |\result| == n;
ensures (\forall int i; 0 <= i && i < n; xs[i] == \result[i]);
pure static seq<int> TakeInt(seq<int> xs, int n) =
0 < n ? seq<int> { head(xs) } + TakeInt(tail(xs), n - 1) : seq<int> { };
// Helper method to extract a color matrix from a set of workers.
requires 0 <= j && j <= |threads|;
requires (\forall* int i; 0 <= i && i < |threads|; threads[i] != null ** Perm(threads[i].color, 1/2));
ensures |\result| == |threads| - j;
ensures (\forall int i; 0 <= i && i < |\result|; \result[i] == threads[j + i].color);
static pure seq<seq<int>> ExtractColorsHelper(seq<Worker> threads, int j) =
j < |threads| ?
seq<seq<int>> { threads[j].color } + ExtractColorsHelper(threads, j + 1) :
seq<seq<int>> { };
// Helper method to extract a pink matrix from a set of workers.
requires 0 <= j && j <= |threads|;
requires (\forall* int i; 0 <= i && i < |threads|; threads[i] != null ** Perm(threads[i].pink, 1/2));
ensures |\result| == |threads| - j;
ensures (\forall int i; 0 <= i && i < |\result|; \result[i] == threads[j + i].pink);
static pure seq<seq<boolean>> ExtractPinkHelper(seq<Worker> threads, int j) =
j < |threads| ?
seq<seq<boolean>> { threads[j].pink } + ExtractPinkHelper(threads, j + 1) :
seq<seq<boolean>> { };
// Gives a matrix representation of the colors of the given set of workers.
requires (\forall* int i; 0 <= i && i < |threads|; threads[i] != null ** Perm(threads[i].color, 1/2));
ensures |\result| == |threads|;
ensures (\forall int i; 0 <= i && i < |threads|; \result[i] == threads[i].color);
static pure seq<seq<int>> ExtractColors(seq<Worker> threads) = ExtractColorsHelper(threads, 0);
// Gives a matrix representation of the pink colorings of the given set of workers.
requires (\forall* int i; 0 <= i && i < |threads|; threads[i] != null ** Perm(threads[i].pink, 1/2));
ensures |\result| == |threads|;
ensures (\forall int i; 0 <= i && i < |threads|; \result[i] == threads[i].pink);
static pure seq<seq<boolean>> ExtractPink(seq<Worker> threads) = ExtractPinkHelper(threads, 0);
/* ** Standard properties of paths */
requires AdjacencyMatrix(N, G) && Path(N, G, x, y, P);
ensures \result && ExPath(N, G, x, y, |P|);
pure static boolean lemma_expath(int N, seq<seq<boolean>> G, int x, int y, seq<int> P) = true;
requires AdjacencyMatrix(N, G) && Path(N, G, x, y, P);
ensures ExPath(N, G, x, y, |P|);
static void lemma_expath_m(int N, seq<seq<boolean>> G, int x, int y, seq<int> P) {
assert lemma_expath(N, G, x, y, P);
}
requires AdjacencyMatrix(N, G);
requires Path(N, G, s, t, P);
requires Path(N, G, t, u, Q);
ensures \result && Path(N, G, s, u, P + tail(Q));
pure static boolean lemma_path_trans(int N, seq<seq<boolean>> G, int s, int t, int u, seq<int> P, seq<int> Q) = true;
requires AdjacencyMatrix(N, G);
requires ExPath(N, G, s, t, len1);
requires ExPath(N, G, t, u, len2);
ensures \result && ExPath(N, G, s, u, len1 + len2 - 1);
pure static boolean lemma_expath_trans(int N, seq<seq<boolean>> G, int s, int t, int u, int len1, int len2) =
(\forall seq<int> P; Path(N, G, s, t, P);
(\forall seq<int> Q; Path(N, G, t, u, Q);
lemma_path_trans(N, G, s, t, u, P, Q)));
requires AdjacencyMatrix(N, G);
requires ExPath(N, G, s, t, len1);
requires ExPath(N, G, t, u, len2);
ensures ExPath(N, G, s, u, len1 + len2 - 1);
static void lemma_expath_trans_m(int N, seq<seq<boolean>> G, int s, int t, int u, int len1, int len2) {
assert lemma_expath_trans(N, G, s, t, u, len1, len2);
}
requires AdjacencyMatrix(N, G);
requires 0 <= s && s < N;
requires 0 <= t && t < N;
requires G[s][t];
ensures \result && ExPath(N, G, s, t, 2);
pure static boolean lemma_edge_path(int N, seq<seq<boolean>> G, int s, int t) =
Path(N, G, s, t, seq<int> { s, t });
requires AdjacencyMatrix(N, G);
requires 0 <= s && s < N;
requires 0 <= t && t < N;
requires G[s][t];
ensures ExPath(N, G, s, t, 2);
static void lemma_edge_path_m(int N, seq<seq<boolean>> G, int s, int t) {
assert lemma_edge_path(N, G, s, t);
}
requires AdjacencyMatrix(N, G) && |color| == N && |pink| == N && |red| == N;
requires 1 < |P| && Path(N, G, P);
requires SpecialPath(N, G, P, color, pink, red);
ensures ExSpecialPath(N, G, color, pink, red);
ensures \result;
static pure boolean lemma_specialpath_ex(int N, seq<seq<boolean>> G, seq<int> color, seq<boolean> pink, seq<boolean> red, seq<int> P) = true;
/* ** Auxiliary lemmas */
// Lemma: If `P` is an `(x,y)`-path, `x` is blue and all successors of blue states are blue, then `y` must consequently also be blue.
requires AdjacencyMatrix(N, G) && |C1| == N;
requires Path(N, G, x, y, P);
requires C1[x] == blue();
requires (\forall int v; 0 <= v && v < N && C1[v] == blue(); (\forall int w; 0 <= w && w < N && G[v][w]; C1[w] == blue())); // `Next(Blue, Blue)`
ensures \result && C1[y] == blue();
pure static boolean lemma_reachblue(int N, seq<seq<boolean>> G, seq<int> C1, int x, int y, seq<int> P) =
1 < |P| ==> lemma_reachblue(N, G, C1, P[1], y, tail(P));
// Lemma: If there is an edge between nodes `s` and `t`, then all paths from any cyan state `v` to `s` can be extended to a `(v,t)`-path.
requires AdjacencyMatrix(N, G) && |C1| == N;
requires 0 <= s && s < N && 0 <= t && t < N && G[s][t];
requires (\forall int v; 0 <= v && v < N && C1[v] == cyan(); ExPath(N, G, v, s, 1));
ensures (\forall int v; 0 <= v && v < N && C1[v] == cyan(); ExPath(N, G, v, t, 1));
ensures \result;
pure static boolean lemma_nextcyan(int N, seq<seq<boolean>> G, seq<int> C1, int s, int t) =
(\forall int v; 0 <= v && v < N && C1[v] == cyan();
lemma_edge_path(N, G, s, t) && lemma_expath_trans(N, G, v, s, t, 1, 1));
requires AdjacencyMatrix(N, G) && |C1| == N;
requires 0 <= s && s < N && 0 <= t && t < N && G[s][t];
requires (\forall int v; 0 <= v && v < N && C1[v] == cyan(); ExPath(N, G, v, s, 1));
ensures (\forall int v; 0 <= v && v < N && C1[v] == cyan(); ExPath(N, G, v, t, 1));
static void lemma_nextcyan_m(int N, seq<seq<boolean>> G, seq<int> C1, int s, int t) {
assert lemma_nextcyan(N, G, C1, s, t);
}
requires AdjacencyMatrix(N, G) && |C1| == N;
requires 1 < |P| && Path(N, G, s, t, P);
requires C1[t] == cyan();
requires (\forall int v; 0 <= v && v < N && C1[v] == cyan(); ExPath(N, G, v, s, 1));
ensures ExPath(N, G, s, s, 2);
ensures \result;
pure static boolean lemma_cyclefoundhere(int N, seq<seq<boolean>> G, seq<int> C1, int s, int t, seq<int> P) =
lemma_expath_trans(N, G, s, t, s, 2, 1);
requires AdjacencyMatrix(N, G) && |C1| == N;
requires 1 < |P| && Path(N, G, s, t, P);
requires C1[t] == cyan();
requires (\forall int v; 0 <= v && v < N && C1[v] == cyan(); ExPath(N, G, v, s, 1));
ensures ExPath(N, G, s, s, 2);
static void lemma_cyclefoundhere_m(int N, seq<seq<boolean>> G, seq<int> C1, int s, int t, seq<int> P) {
assert lemma_cyclefoundhere(N, G, C1, s, t, P);
}
/* ** Color invariants */
// `Next(Blue, Blue ∪ Cyan ∪ Red)`
static inline pure boolean Invariant1a(int N, seq<seq<boolean>> G, seq<int> color, seq<boolean> red) =
(\forall int v; 0 <= v && v < N && color[v] == blue();
(\forall int w; 0 <= w && w < N && G[v][w];
color[w] == blue() || color[w] == cyan() || red[w]));
// `Next(Pink, Blue ∪ Cyan ∪ Red)`
static inline pure boolean Invariant1b(int N, seq<seq<boolean>> G, seq<int> color, seq<boolean> pink, seq<boolean> red) =
(\forall int v; 0 <= v && v < N && pink[v];
(\forall int w; 0 <= w && w < N && G[v][w];
color[w] == blue() || color[w] == cyan() || red[w]));
// `A ∩ Blue ⊆ Red`
static inline pure boolean Invariant3(int N, seq<seq<boolean>> G, seq<boolean> A, seq<int> color, seq<boolean> red) =
(\forall int v; 0 <= v && v < N && A[v] && color[v] == blue(); red[v]);