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Dot.dfy
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Dot.dfy
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// ============================
// Dependent Object Types (DOT)
// ============================
// ---------
// Utilities
// ---------
// ### List ###
datatype list<A> = Nil | Cons(head: A, tail: list<A>);
// ### Pair ###
datatype pair<A, B> = P(fst: A, snd: B);
// ### Option ###
datatype option<A> = None | Some(get: A);
function seq2lst<A>(s: seq<A>): list<A>
{
if (s == []) then Nil
else Cons(s[0], seq2lst(s[1..]))
}
function lst2seq<A>(lst: list<A>): seq<A>
{
match lst
case Nil => []
case Cons(head, tail) => [head] + lst2seq(tail)
}
function max(s: seq<int>, start: int): int
ensures max(s, start)>=start;
ensures forall x :: x in s ==> x<=max(s, start);
{
if (s == []) then start
else if (s[0] <= start) then max(s[1..], start)
else max(s[1..], s[0])
}
function fresh_from(ids: seq<int>): int
ensures fresh_from(ids) !in ids;
ensures fresh_from(ids)>0;
{
max(ids, 0)+1
}
// ------
// Syntax
// ------
datatype tp = tp_sel(p: tm, L: nat, concrete: bool) | tp_rfn(base_tp: tp, self: nat, decls: list<decl>) | tp_and(and1: tp, and2: tp) | tp_or(or1: tp, or2: tp) | tp_top | tp_bot;
datatype tm = tm_var(x: nat) | tm_new(y: nat, Tc: tp, init: list<def>, t': tm) | tm_sel(t: tm, l: nat) | tm_msel(o: tm, m: nat, a: tm) | tm_loc(loc: nat);
datatype decl = decl_tp(L: nat, S: tp, U: tp, concrete: bool) | decl_tm(l: nat, T: tp) | decl_mt(m: nat, P: tp, R: tp);
datatype def = def_tm(l: nat, t: tm) | def_mt(m: nat, param: nat, body: tm);
datatype decls = decls_fin(decls: list<decl>) | decls_bot;
predicate path(t: tm)
{
t.tm_loc? || t.tm_var? || (t.tm_sel? && path(t.t))
}
predicate is_concrete(T: tp)
{
(T.tp_sel? && T.concrete && path(T.p)) ||
(T.tp_rfn? && is_concrete(T.base_tp)) ||
(T.tp_and? && is_concrete(T.and1) && is_concrete(T.and2)) ||
T.tp_top?
}
function decl_label(d: decl): nat
{
match d
case decl_tp(L, S, U, concrete) => L
case decl_tm(l, T) => l
case decl_mt(m, P, R) => m
}
// ---------------------
// Operational Semantics
// ---------------------
// ### Values ###
predicate value(t: tm)
{
t.tm_loc?
}
predicate syn_value(t: tm)
{
t.tm_loc? || t.tm_var?
}
// ### Store ###
datatype store = Store(m: seq<pair<tp, seq<def>>>);
function store_lookup(loc: nat, s: store): seq<def>
requires loc < |s.m|;
{
s.m[loc].snd
}
function store_lookup_type(loc: nat, s: store): tp
requires loc < |s.m|;
{
s.m[loc].fst
}
function def_method_lookup(m: nat, defs: seq<def>): option<pair<int, tm>>
ensures def_method_lookup(m, defs).Some? ==> def_method_lookup(m, defs).get.fst>=0;
{
if (exists i :: 0 <= i < |defs| && defs[i].def_mt? && defs[i].m==m)
then (var i :| 0 <= i < |defs| && defs[i].def_mt? && defs[i].m==m; Some(P(defs[i].param, defs[i].body)))
else None
}
function def_field_lookup(l: nat, defs: seq<def>): option<tm>
{
if (exists i :: 0 <= i < |defs| && defs[i].def_tm? && defs[i].l==l)
then (var i :| 0 <= i < |defs| && defs[i].def_tm? && defs[i].l==l; Some(defs[i].t))
else None
}
// ### Size ###
function tm_size(t: tm): nat
ensures t.tm_new? ==> tm_size(t)==1+tp_size(t.Tc)+defs_size(t.init)+tm_size(t.t');
ensures t.tm_new? ==> tm_size(t)>tp_size(t.Tc);
ensures t.tm_new? ==> tm_size(t)>defs_size(t.init);
ensures t.tm_new? ==> tm_size(t)>tm_size(t.t');
ensures t.tm_sel? ==> tm_size(t)==1+tm_size(t.t);
ensures t.tm_msel? ==> tm_size(t)==1+tm_size(t.o)+tm_size(t.a);
ensures t.tm_msel? ==> tm_size(t)>tm_size(t.o);
{
match t
case tm_var(x') => 1
case tm_new(y, Tc, init, t1) => 1+tp_size(Tc)+defs_size(init)+tm_size(t1)
case tm_sel(t1, l) => 1+tm_size(t1)
case tm_msel(o, m, a) => 1+tm_size(o)+tm_size(a)
case tm_loc(loc) => 1
}
function tp_size(T: tp): nat
ensures T.tp_sel? ==> tp_size(T)==1+tm_size(T.p);
ensures T.tp_sel? ==> tp_size(T)>tm_size(T.p);
ensures T.tp_rfn? ==> tp_size(T)==1+tp_size(T.base_tp)+decls_size(T.decls);
ensures T.tp_rfn? ==> tp_size(T)>tp_size(T.base_tp);
ensures T.tp_rfn? ==> tp_size(T)>decls_size(T.decls);
ensures T.tp_and? ==> tp_size(T)==1+tp_size(T.and1)+tp_size(T.and2);
ensures T.tp_and? ==> tp_size(T)>tp_size(T.and1);
ensures T.tp_or? ==> tp_size(T)==1+tp_size(T.or1)+tp_size(T.or2);
ensures T.tp_or? ==> tp_size(T)>tp_size(T.or1);
{
match T
case tp_sel(p, L, concrete) => 1+tm_size(p)
case tp_rfn(base_tp, self, decls) => 1+tp_size(base_tp)+decls_size(decls)
case tp_and(and1, and2) => 1+tp_size(and1)+tp_size(and2)
case tp_or(or1, or2) => 1+tp_size(or1)+tp_size(or2)
case tp_top => 1
case tp_bot => 1
}
function def_size(d: def): nat
ensures d.def_tm? ==> def_size(d)==1+tm_size(d.t);
ensures d.def_tm? ==> def_size(d)>tm_size(d.t);
ensures d.def_mt? ==> def_size(d)==1+tm_size(d.body);
ensures d.def_mt? ==> def_size(d)>tm_size(d.body);
{
match d
case def_tm(l, t1) => 1+tm_size(t1)
case def_mt(m, param, body) => 1+tm_size(body)
}
function decl_size(d: decl): nat
ensures d.decl_tp? ==> decl_size(d)==1+tp_size(d.S)+tp_size(d.U);
ensures d.decl_tp? ==> decl_size(d)>tp_size(d.S);
ensures d.decl_tp? ==> decl_size(d)>tp_size(d.U);
ensures d.decl_tm? ==> decl_size(d)==1+tp_size(d.T);
ensures d.decl_tm? ==> decl_size(d)>tp_size(d.T);
ensures d.decl_mt? ==> decl_size(d)==1+tp_size(d.P)+tp_size(d.R);
ensures d.decl_mt? ==> decl_size(d)>tp_size(d.P);
ensures d.decl_mt? ==> decl_size(d)>tp_size(d.R);
{
match d
case decl_tp(L, S, U, concrete) => 1+tp_size(S)+tp_size(U)
case decl_tm(l, T) => 1+tp_size(T)
case decl_mt(m, P, R) => 1+tp_size(P)+tp_size(R)
}
function defs_size(defs: list<def>): nat
ensures defs.Cons? ==> defs_size(defs)==1+def_size(defs.head)+defs_size(defs.tail);
ensures defs.Cons? ==> defs_size(defs)>def_size(defs.head);
{
match defs
case Nil => 1
case Cons(head, tail) => 1+def_size(head)+defs_size(tail)
}
function decls_size(decls: list<decl>): nat
ensures decls.Cons? ==> decls_size(decls)==1+decl_size(decls.head)+decls_size(decls.tail);
ensures decls.Cons? ==> decls_size(decls)>decl_size(decls.head);
{
match decls
case Nil => 1
case Cons(head, tail) => 1+decl_size(head)+decls_size(tail)
}
// ### Substitution ###
function tm_subst(x: nat, v: tm, t: tm): tm
decreases tm_size(t), t;
ensures v.tm_var? ==> tm_size(t)==tm_size(tm_subst(x, v, t));
ensures path(v) && path(t) ==> path(tm_subst(x, v, t));
ensures !tm_fn(x, t) ==> tm_subst(x, v, t)==t;
{
match t
case tm_var(x') => if x'==x then v else t
case tm_new(y, Tc, init, t1) =>
if !tm_fn(x, t) then t else
if y==x then tm_new(y, tp_subst(x, v, Tc), init, t1) else
var y' := fresh_from([x]+tm_vars(v)+tm_vars(t));
var init' := defs_subst(y, tm_var(y'), init);
var t1' := tm_subst(y, tm_var(y'), t1);
tm_new(y', tp_subst(x, v, Tc), defs_subst(x, v, init'), tm_subst(x, v, t1'))
case tm_sel(t1, l) => tm_sel(tm_subst(x, v, t1), l)
case tm_msel(o, m, a) => tm_msel(tm_subst(x, v, o), m, tm_subst(x, v, a))
case tm_loc(loc) => t
}
function tp_subst(x: nat, v: tm, T: tp): tp
decreases tp_size(T), T;
ensures v.tm_var? ==> tp_size(T)==tp_size(tp_subst(x, v, T));
ensures !tp_fn(x, T) ==> tp_subst(x, v, T)==T;
{
match T
case tp_sel(p, L, concrete) => tp_sel(tm_subst(x, v, p), L, concrete)
case tp_rfn(base_tp, self, decls) =>
if !tp_fn(x, T) then T else
if self==x then tp_rfn(tp_subst(x, v, base_tp), self, decls) else
var self' := fresh_from([x]+tm_vars(v)+tp_vars(T));
var decls' := decls_subst(self, tm_var(self'), decls);
tp_rfn(tp_subst(x, v, base_tp), self', decls_subst(x, v, decls'))
case tp_and(and1, and2) => tp_and(tp_subst(x, v, and1), tp_subst(x, v, and2))
case tp_or(or1, or2) => tp_or(tp_subst(x, v, or1), tp_subst(x, v, or2))
case tp_top => T
case tp_bot => T
}
function def_subst(x: nat, v: tm, d: def): def
decreases def_size(d), d;
ensures v.tm_var? ==> def_size(d)==def_size(def_subst(x, v, d));
ensures !def_fn(x, d) ==> def_subst(x, v, d)==d;
{
match d
case def_tm(l, t1) => def_tm(l, tm_subst(x, v, t1))
case def_mt(m, param, body) =>
if !def_fn(x, d) then d else
if param==x then d else
var param' := fresh_from([x]+tm_vars(v)+def_vars(d));
var body' := tm_subst(param, tm_var(param'), body);
def_mt(m, param', tm_subst(x, v, body'))
}
function decl_subst(x: nat, v: tm, d: decl): decl
decreases decl_size(d), d;
ensures v.tm_var? ==> decl_size(d)==decl_size(decl_subst(x, v, d));
ensures !decl_fn(x, d) ==> decl_subst(x, v, d)==d;
{
match d
case decl_tp(L, S, U, concrete) => decl_tp(L, tp_subst(x, v, S), tp_subst(x, v, U), concrete)
case decl_tm(l, T) => decl_tm(l, tp_subst(x, v, T))
case decl_mt(m, P, R) => decl_mt(m, tp_subst(x, v, P), tp_subst(x, v, R))
}
function defs_subst(x: nat, v: tm, defs: list<def>): list<def>
decreases defs_size(defs), defs;
ensures v.tm_var? ==> defs_size(defs)==defs_size(defs_subst(x, v, defs));
ensures !defs_fn(x, defs) ==> defs_subst(x, v, defs)==defs;
{
match defs
case Nil => Nil
case Cons(head, tail) => Cons(def_subst(x, v, head), defs_subst(x, v, tail))
}
function decls_subst(x: nat, v: tm, decls: list<decl>): list<decl>
decreases decls_size(decls), decls;
ensures v.tm_var? ==> decls_size(decls)==decls_size(decls_subst(x, v, decls));
ensures !decls_fn(x, decls) ==> decls_subst(x, v, decls)==decls;
{
match decls
case Nil => Nil
case Cons(head, tail) => Cons(decl_subst(x, v, head), decls_subst(x, v, tail))
}
// ### Free variables ###
// fn(x, A) <==> x appears free in A
predicate tm_fn(x: nat, t: tm)
ensures tm_fn(x, t) ==> x in tm_vars(t);
decreases t;
{
match t
case tm_var(x') => x'==x
case tm_new(y, Tc, init, t') => tp_fn(x, Tc) || (y!=x && (defs_fn(x, init) || tm_fn(x, t')))
case tm_sel(t1, l) => tm_fn(x, t1)
case tm_msel(o, m, a) => tm_fn(x, o) || tm_fn(x, a)
case tm_loc(loc) => false
}
predicate tp_fn(x: nat, T: tp)
ensures tp_fn(x, T) ==> x in tp_vars(T);
decreases T;
{
match T
case tp_sel(p, L, concrete) => tm_fn(x, p)
case tp_rfn(base_tp, self, decls) => tp_fn(x, base_tp) || (self!=x && decls_fn(x, decls))
case tp_and(and1, and2) => tp_fn(x, and1) || tp_fn(x, and2)
case tp_or(or1, or2) => tp_fn(x, or1) || tp_fn(x, or2)
case tp_top => false
case tp_bot => false
}
predicate def_fn(x: nat, d: def)
ensures def_fn(x, d) ==> x in def_vars(d);
decreases d;
{
match d
case def_tm(l, t1) => tm_fn(x, t1)
case def_mt(m, param, body) => param!=x && tm_fn(x, body)
}
predicate decl_fn(x: nat, d: decl)
ensures decl_fn(x, d) ==> x in decl_vars(d);
decreases d;
{
match d
case decl_tp(L, S, U, concrete) => tp_fn(x, S) || tp_fn(x, U)
case decl_tm(l, T) => tp_fn(x, T)
case decl_mt(m, P, R) => tp_fn(x, P) || tp_fn(x, R)
}
predicate defs_fn(x: nat, defs: list<def>)
ensures defs_fn(x, defs) ==> x in defs_vars(defs);
decreases defs;
{
defs.Cons? && (def_fn(x, defs.head) || defs_fn(x, defs.tail))
}
predicate decls_fn(x: nat, decls: list<decl>)
ensures decls_fn(x, decls) ==> x in decls_vars(decls);
decreases decls;
{
decls.Cons? && (decl_fn(x, decls.head) || decls_fn(x, decls.tail))
}
// ### Variables ###
function tm_vars(t: tm): seq<int>
ensures forall x :: x in tm_vars(t) ==> x>=0;
{
match t
case tm_var(x') => [x']
case tm_new(y, Tc, init, t') => [y]+tp_vars(Tc)+defs_vars(init)+tm_vars(t')
case tm_sel(t1, l) => tm_vars(t1)
case tm_msel(o, m, a) => tm_vars(o)+tm_vars(a)
case tm_loc(loc) => []
}
function tp_vars(T: tp): seq<int>
ensures forall x :: x in tp_vars(T) ==> x>=0;
{
match T
case tp_sel(p, L, concrete) => tm_vars(p)
case tp_rfn(base_tp, self, decls) => tp_vars(base_tp)+[self]+decls_vars(decls)
case tp_and(and1, and2) => tp_vars(and1)+tp_vars(and2)
case tp_or(or1, or2) => tp_vars(or1)+tp_vars(or2)
case tp_top => []
case tp_bot => []
}
function def_vars(d: def): seq<int>
ensures forall x :: x in def_vars(d) ==> x>=0;
{
match d
case def_tm(l, t1) => tm_vars(t1)
case def_mt(m, param, body) => [param]+tm_vars(body)
}
function decl_vars(d: decl): seq<int>
ensures forall x :: x in decl_vars(d) ==> x>=0;
{
match d
case decl_tp(L, S, U, concrete) => tp_vars(S)+tp_vars(U)
case decl_tm(l, T) => tp_vars(T)
case decl_mt(m, P, R) => tp_vars(P)+tp_vars(R)
}
function defs_vars(defs: list<def>): seq<int>
ensures forall x :: x in defs_vars(defs) ==> x>=0;
ensures defs.Cons? ==> defs_vars(defs)==def_vars(defs.head)+defs_vars(defs.tail);
{
match defs
case Nil => []
case Cons(head, tail) => def_vars(head)+defs_vars(tail)
}
function decls_vars(decls: list<decl>): seq<int>
ensures forall x :: x in decls_vars(decls) ==> x>=0;
ensures decls.Cons? ==> decls_vars(decls)==decl_vars(decls.head)+decls_vars(decls.tail);
{
match decls
case Nil => []
case Cons(head, tail) => decl_vars(head)+decls_vars(tail)
}
// ### Reduction ###
function step(t: tm, s: store): option<pair<tm, store>>
{
/* msel */
if (t.tm_msel? && t.o.tm_loc? && value(t.a) && t.o.loc < |s.m| &&
def_method_lookup(t.m, store_lookup(t.o.loc, s)).Some?)
then Some(P(tm_subst(def_method_lookup(t.m, store_lookup(t.o.loc, s)).get.fst,
t.a,
def_method_lookup(t.m, store_lookup(t.o.loc, s)).get.snd),
s))
/* msel1 */
else if (t.tm_msel? && step(t.o, s).Some?)
then Some(P(tm_msel(step(t.o, s).get.fst, t.m, t.a), step(t.o, s).get.snd))
/* msel2 */
else if (t.tm_msel? && value(t.o) && step(t.a, s).Some?)
then Some(P(tm_msel(t.o, t.m, step(t.a, s).get.fst), step(t.a, s).get.snd))
/* sel */
else if (t.tm_sel? && t.t.tm_loc? && t.t.loc < |s.m| &&
def_field_lookup(t.l, store_lookup(t.t.loc, s)).Some?)
then Some(P(def_field_lookup(t.l, store_lookup(t.t.loc, s)).get, s))
/* sel1 */
else if (t.tm_sel? && step(t.t, s).Some?)
then Some(P(tm_sel(step(t.t, s).get.fst, t.l), step(t.t, s).get.snd))
/* new */
else if (t.tm_new?)
then Some(P(tm_subst(t.y, tm_loc(|s.m|), t.t'),
Store(s.m+[P(t.Tc, lst2seq(defs_subst(t.y, tm_loc(|s.m|), t.init)))])))
else None
}
predicate irred(t: tm, s: store)
{
step(t, s).None?
}
// ### Multi-steps ###
predicate mstep(t: tm, s: store, t': tm, s': store, n: nat)
decreases n;
{
if (n==0) then t==t' && s==s'
else step(t, s).Some? && mstep(step(t, s).get.fst, step(t, s).get.snd, t', s', n-1)
}
// ### Properties of Operational Semantics ###
ghost method lemma_value__irred(t: tm, s: store)
requires value(t);
ensures irred(t, s);
{
}
ghost method lemma_mstep_trans(t1: tm, s1: store, t2: tm, s2: store, t3: tm, s3: store, n12: nat, n23: nat)
requires mstep(t1, s1, t2, s2, n12);
requires mstep(t2, s2, t3, s3, n23);
ensures mstep(t1, s1, t3, s3, n12+n23);
decreases n12+n23;
{
if (n12>0) {
lemma_mstep_trans(step(t1, s1).get.fst, step(t1, s1).get.snd, t2, s2, t3, s3, n12-1, n23);
} else if (n23>0) {
lemma_mstep_trans(step(t1, s1).get.fst, step(t1, s1).get.snd, step(t2, s2).get.fst, step(t2, s2).get.snd, t3, s3, n12, n23-1);
}
}
ghost method lemma_mstep_trans'(t1: tm, s1: store, t2: tm, s2: store, t3: tm, s3: store, n12: nat, n13: nat)
requires n12 <= n13;
requires mstep(t1, s1, t2, s2, n12);
requires mstep(t1, s1, t3, s3, n13);
ensures mstep(t2, s2, t3, s3, n13-n12);
decreases n12;
{
if (n12>0 && n13>0) {
lemma_mstep_trans'(step(t1, s1).get.fst, step(t1, s1).get.snd, t2, s2, t3, s3, n12-1, n13-1);
}
}
// ### Congruence Lemmas ###
ghost method lemma_mstep_sel(o: tm, l: nat, o': tm, s: store, s': store, oi: nat)
requires mstep(o, s, o', s', oi);
ensures mstep(tm_sel(o, l), s, tm_sel(o', l), s', oi);
decreases oi;
{
if (oi>0) {
lemma_mstep_sel(step(o, s).get.fst, l, o', step(o, s).get.snd, s', oi-1);
}
}
ghost method lemma_sel_irred__o_mstep_irred(o: tm, l: nat, t': tm, s: store, s': store, i: nat) returns (o': tm, so': store, oi: nat)
requires mstep(tm_sel(o, l), s, t', s', i);
requires irred(t', s');
ensures oi<=i && mstep(o, s, o', so', oi) && irred(o', so');
decreases i;
{
if (irred(o, s)) {
o' := o;
so' := s;
oi := 0;
} else {
assert step(o, s).Some?;
lemma_mstep_trans'(tm_sel(o, l), s, tm_sel(step(o, s).get.fst, l), step(o, s).get.snd, t', s', 1, i);
var o'', so'', oi' := lemma_sel_irred__o_mstep_irred(step(o, s).get.fst, l, t', step(o, s).get.snd, s', i-1);
lemma_mstep_trans(o, s, step(o, s).get.fst, step(o, s).get.snd, o'', so'', 1, oi');
o' := o'';
so' := so'';
oi := oi'+1;
}
}
// -----------
// Type System
// -----------
// ### Context ###
datatype context = Context(m: seq<pair<int,tp>>);
function context_extend(ctx: context, x: nat, T: tp): context
requires context_lookup(ctx, x).None?;
{
Context([P(x, T)]+ctx.m)
}
function context_lookup(ctx: context, x: nat): option<tp>
decreases |ctx.m|;
{
if (|ctx.m|==0) then None
else if (ctx.m[0].fst==x) then Some(ctx.m[0].snd)
else context_lookup(Context(ctx.m[1..]), x)
}
function dom(ctx: context): seq<int>
ensures forall x:nat :: x !in dom(ctx) ==> context_lookup(ctx, x).None?;
ensures forall x:nat :: x in dom(ctx) ==> context_lookup(ctx, x).Some?;
decreases |ctx.m|;
{
if (ctx.m==[]) then [] else [ctx.m[0].fst]+dom(Context(ctx.m[1..]))
}
function fresh_in_context(ctx: context): nat
ensures context_lookup(ctx, fresh_in_context(ctx)).None?;
{
fresh_from(dom(ctx))
}
// ### Orderings ###
predicate decl_lt(d1: decl, d2: decl)
{
match d1
case decl_tp(L1, S1, U1, concrete1) =>
(match d2
case decl_tp(L2, S2, U2, concrete2) =>
(concrete1 && !concrete2) || (concrete1==concrete2 && L1<L2)
case decl_tm(l2, T2) => true
case decl_mt(m2, P2, R2) => true)
case decl_tm(l1, T1) =>
(match d2
case decl_tp(L2, S2, U2, concrete2) => false
case decl_tm(l2, T2) => l1<l2
case decl_mt(m2, P2, R2) => true)
case decl_mt(m1, P1, R1) =>
(match d2
case decl_tp(L2, S2, U2, concrete2) => false
case decl_tm(l2, T2) => false
case decl_mt(m2, P2, R2) => m1<m2)
}
predicate decl_eq(d1: decl, d2: decl)
ensures d1==d2 ==> decl_eq(d1, d2);
ensures decl_eq(d1, d2) ==> decl_label(d1)==decl_label(d2);
ensures decl_eq(d1, d2) ==> d1.decl_tp?==d2.decl_tp?;
ensures decl_eq(d1, d2) ==> d1.decl_tm?==d2.decl_tm?;
ensures decl_eq(d1, d2) ==> d1.decl_mt?==d2.decl_mt?;
ensures decl_eq(d1, d2) ==> !decl_lt(d1, d2) && !decl_lt(d2, d1);
{
match d1
case decl_tp(L1, S1, U1, concrete1) => d2.decl_tp? && d2.concrete==concrete1 && d2.L==L1
case decl_tm(l1, T1) => d2.decl_tm? && d2.l==l1
case decl_mt(m1, P1, R1) => d2.decl_mt? && d2.m==m1
}
predicate decl_le(d1: decl, d2: decl)
ensures d1==d2 ==> decl_le(d1, d2);
{
decl_lt(d1, d2) || decl_eq(d1, d2)
}
predicate decl_seq_sorted(s: seq<decl>)
{
forall m,n :: 0 <= m < n < |s| ==> decl_le(s[m], s[n])
}
function decl_seq_merge(s1: seq<decl>, s2: seq<decl>): seq<decl>
{
if (s1 == []) then s2
else if (s2 == []) then s1
else if (decl_le(s1[0], s2[0])) then [s1[0]]+decl_seq_merge(s1[1..], s2)
else [s2[0]]+decl_seq_merge(s1, s2[1..])
}
function decl_seq_sort(s: seq<decl>): seq<decl>
{
if (s == []) then s else
var i: nat := (|s|-1)/2;
decl_seq_merge(decl_seq_sort(s[..i]), decl_seq_sort(s[i+1..]))
}
predicate def_lt(d1: def, d2: def)
{
match d1
case def_tm(l1, t1) =>
(match d2
case def_tm(l2, t2) => l1<l2
case def_mt(m2, param2, body2) => true)
case def_mt(m1, param1, body1) =>
(match d2
case def_tm(l2, t2) => false
case def_mt(m2, param2, body2) => m1<m2)
}
predicate def_eq(d1: def, d2: def)
ensures d1==d2 ==> def_eq(d1, d2);
{
match d1
case def_tm(l1, t1) => d2.def_tm? && d2.l==l1
case def_mt(m1, param1, body1) => d2.def_mt? && d2.m==m1
}
predicate def_le(d1: def, d2: def)
ensures d1==d2 ==> def_le(d1, d2);
{
def_lt(d1, d2) || def_eq(d1, d2)
}
predicate def_seq_sorted(s: seq<def>)
{
forall m,n :: 0 <= m < n < |s| ==> def_le(s[m], s[n])
}
function def_seq_merge(s1: seq<def>, s2: seq<def>): seq<def>
{
if (s1 == []) then s2
else if (s2 == []) then s1
else if (def_le(s1[0], s2[0])) then [s1[0]]+def_seq_merge(s1[1..], s2)
else [s2[0]]+def_seq_merge(s1, s2[1..])
}
function def_seq_sort(s: seq<def>): seq<def>
{
if (s == []) then s else
var i: nat := (|s|-1)/2;
def_seq_merge(def_seq_sort(s[..i]), def_seq_sort(s[i+1..]))
}
// ### Declaration Lattice ###
predicate decl_bot(d: decl)
{
match d
case decl_tp(L, S, U, concrete) => S==tp_top && U==tp_bot
case decl_tm(l, T) => T==tp_bot
case decl_mt(m, S, U) => S==tp_top && U==tp_bot
}
function decl_and(d1: decl, d2: decl): decl
requires decl_eq(d1, d2);
ensures decl_eq(d1, decl_and(d1, d2));
ensures decl_eq(d2, decl_and(d1, d2));
{
match d1
case decl_tp(L, S, U, concrete) => decl_tp(L, tp_or(S, d2.S), tp_and(U, d2.U), concrete)
case decl_tm(l, U) => decl_tm(l, tp_and(U, d2.T))
case decl_mt(m, S, U) => decl_mt(m, tp_or(S, d2.P), tp_and(U, d2.R))
}
function decl_or(d1: decl, d2: decl): decl
requires decl_eq(d1, d2);
ensures decl_eq(d1, decl_or(d1, d2));
ensures decl_eq(d2, decl_or(d1, d2));
{
match d1
case decl_tp(L, S, U, concrete) => decl_tp(L, tp_and(S, d2.S), tp_or(U, d2.U), concrete)
case decl_tm(l, U) => decl_tm(l, tp_or(U, d2.T))
case decl_mt(m, S, U) => decl_mt(m, tp_and(S, d2.P), tp_or(U, d2.R))
}
function decls_fin_and(s1: seq<decl>, s2: seq<decl>): seq<decl>
{
if (s1 == []) then s2
else if (s2 == []) then s1
else if (decl_eq(s1[0], s2[0])) then [decl_and(s1[0], s2[0])]+decls_fin_and(s1[1..], s2[1..])
else if (decl_lt(s1[0], s2[0])) then [s1[0]]+decls_fin_and(s1[1..], s2)
else/*if (decl_lt(s2[0], s1[0])) then*/ [s2[0]]+decls_fin_and(s1, s2[1..])
}
function decls_and(Ds1: decls, Ds2: decls): decls
{
match Ds1
case decls_bot => decls_bot
case decls_fin(s1) =>
(match Ds2
case decls_bot => decls_bot
case decls_fin(s2) => decls_fin(seq2lst(decls_fin_and(lst2seq(s1), lst2seq(s2)))))
}
function decls_fin_or(s1: seq<decl>, s2: seq<decl>): seq<decl>
{
if (s1 == []) then []
else if (s2 == []) then []
else if (decl_eq(s1[0], s2[0])) then [decl_or(s1[0], s2[0])]+decls_fin_and(s1[1..], s2[1..])
else if (decl_lt(s1[0], s2[0])) then decls_fin_or(s1[1..], s2)
else/*if (decl_lt(s2[0], s1[0])) then*/ decls_fin_or(s1, s2[1..])
}
function decls_or(Ds1: decls, Ds2: decls): decls
{
match Ds1
case decls_bot => Ds2
case decls_fin(s1) =>
(match Ds2
case decls_bot => Ds1
case decls_fin(s2) => decls_fin(seq2lst(decls_fin_or(lst2seq(s1), lst2seq(s2)))))
}
// ### Typing-Related Judgments ###
predicate typing(n: nat, ctx: context, s: store, t: tm, T: tp)
decreases n;
{
match t
case tm_var(x) => context_lookup(ctx, x)==Some(T)
case tm_new(y_, Tc, init_, t'_) =>
var y := fresh_from(dom(ctx)+tm_vars(t));
var init := defs_subst(y_, tm_var(y), init_);
var t' := tm_subst(y_, tm_var(y), t'_);
n>0 && is_concrete(Tc) &&
exists Ds:decls :: Ds.decls_fin? &&
wfe_type(n-1, ctx, s, Tc) &&
expansion(n-1, ctx, s, y, Tc, Ds) &&
wf_init(n-1, false, context_extend(ctx, y, Tc), s, lst2seq(Ds.decls), lst2seq(init)) &&
!tp_fn(y, T) &&
exists T' :: typing(n-1, context_extend(ctx, y, Tc), s, t', T') &&
subtype(n-1, context_extend(ctx, y, Tc), s, T', T)
case tm_sel(t, l) =>
n>0 && field_membership(n-1, ctx, s, t, l, T)
case tm_msel(o, m, a) =>
n>0 && exists S, T' :: method_membership(n-1, ctx, s, o, m, S, T) &&
typing(n-1, ctx, s, a, T') &&
subtype(n-1, ctx, s, T', S)
case tm_loc(loc) => loc < |s.m| && store_lookup_type(loc, s)==T
}
predicate wf_init(n: nat, already_in_store: bool, ctx: context, s: store, decls: seq<decl>, defs: seq<def>)
decreases n;
{
var p:nat := fresh_in_context(ctx);
n>0 && forall d :: d in decls ==> (
if (d.decl_tp?) then subtype(n-1, ctx, s, d.S, d.U)
else if (d.decl_tm?) then exists def :: def in defs && def.def_tm? && def.l==d.l && (if already_in_store then value(def.t) else syn_value(def.t)) &&
exists T :: typing(n-1, ctx, s, def.t, T) && subtype(n-1, ctx, s, T, d.T)
else if (d.decl_mt?) then exists def :: def in defs && def.def_mt? && def.m==d.m &&
exists T' :: typing(n-1, context_extend(ctx, p, d.P), s, tm_subst(def.param, tm_var(p), def.body), T') &&
subtype(n-1, context_extend(ctx, p, d.P), s, T', d.R)
else false)
}
predicate wf_decl(n: nat, ctx: context, s: store, d: decl)
decreases n;
{
match d
case decl_tp(L, S, U, concrete) => n>0 && wf_type(n-1, ctx, s, S) && wf_type(n-1, ctx, s, U) &&
(!concrete || (S==tp_bot && is_concrete(U)))
case decl_tm(l, T) => n>0 && wf_type(n-1, ctx, s, T)
case decl_mt(m, S, T) => n>0 && wf_type(n-1, ctx, s, S) && wf_type(n-1, ctx, s, T)
}
predicate wf_decls(n: nat, ctx: context, s: store, Ds: seq<decl>)
decreases n;
{
forall d :: d in Ds ==> n>0 && wf_decl(n-1, ctx, s, d)
}
predicate wf_type(n: nat, ctx: context, s: store, T: tp)
decreases n;
{
match T
case tp_rfn(T', z_, Ds_) =>
var z:= fresh_in_context(ctx);
var Ds := decls_subst(z_, tm_var(z), Ds_);
n>0 && wf_type(n-1, ctx, s, T') && wf_decls(n-1, context_extend(ctx, z, T), s, lst2seq(Ds))
case tp_sel(p, L, concrete) =>
path(p) &&
n>0 && exists S, U :: type_membership(n-1, ctx, s, p, L, concrete, S, U)
case tp_and(T1, T2) => n>0 && wf_type(n-1, ctx, s, T1) && wf_type(n-1, ctx, s, T2)
case tp_or(T1, T2) => n>0 && wf_type(n-1, ctx, s, T1) && wf_type(n-1, ctx, s, T2)
case tp_top => true
case tp_bot => true
}
predicate wfe_type(n: nat, ctx: context, s: store, T: tp)
decreases n;
{
n>0 && wf_type(n-1, ctx, s, T) && exists Ds :: expansion(n-1, ctx, s, 0, T, Ds)
}
predicate membership(n: nat, ctx: context, s: store, t: tm, l: nat, d: decl)
decreases n;
{
var z:nat := fresh_in_context(ctx);
decl_label(d)==l &&
n>0 && exists T, T' :: typing(n-1, ctx, s, t, T') &&
subtype(n-1, ctx, s, T', T) &&
exists Ds ::
expansion(n-1, ctx, s, z, T, Ds) &&
((Ds.decls_fin? &&
((path(t) && T'==T && exists d' :: d' in lst2seq(Ds.decls) && d==decl_subst(z, t, d')) ||
(!path(t) && d in lst2seq(Ds.decls) && !decl_fn(z, d)))) ||
(Ds.decls_bot? && decl_bot(d)))
}
predicate field_membership(n: nat, ctx: context, s: store, t: tm, l: nat, T: tp)
decreases n;
{
n>0 && membership(n-1, ctx, s, t, l, decl_tm(l, T))
}
predicate method_membership(n: nat, ctx: context, s: store, t: tm, m: nat, P: tp, R: tp)
decreases n;
{
n>0 && exists d :: membership(n-1, ctx, s, t, m, d) &&
d.decl_mt? && d.m==m && d.P==P && d.R==R
}
predicate type_membership(n: nat, ctx: context, s: store, t: tm, L: nat, concrete: bool, S: tp, U: tp)
decreases n;
{
n>0 && exists d :: membership(n-1, ctx, s, t, L, d) &&
d.decl_tp? && d.L==L && d.concrete==concrete && d.S==S && d.U==U
}
predicate m_decl_seq_sorted(m: seq<pair<tp, decls>>)
{
forall p :: p in m && p.snd.decls_fin? ==> decl_seq_sorted(lst2seq(p.snd.decls))
}
function lookup<K,V>(k: K, m: seq<pair<K,V>>): option<V>
{
if (exists v :: P(k,v) in m)
then (var v :| P(k, v) in m; Some(v))
else None
}
predicate expansion(n: nat, ctx: context, s: store, z: nat, T: tp, Ds: decls)
decreases n;
{
n>0 && expansion_iter(n-1, [], ctx, s, z, T, Ds)
}
predicate expansion_iter(n: nat, m: seq<pair<tp, decls>>, ctx: context, s: store, z: nat, T: tp, Ds: decls)
decreases n;
{
match T
case tp_rfn(T', z', Ds') =>
n>0 &&
exists DsT' :: expansion_iter(n-1, m, ctx, s, z, T', DsT') &&
exists rfn_decls :: rfn_decls==decl_seq_sort(lst2seq(decls_subst(z', tm_var(z), Ds'))) &&
Ds==decls_and(decls_fin(seq2lst(rfn_decls)), DsT')
case tp_sel(p, L, concrete) =>
(lookup(T, m).Some? && lookup(T, m).get==Ds) ||
(n>0 && lookup(T, m).None? && exists S, U :: type_membership(n-1, ctx, s, p, L, concrete, S, U) &&
expansion_fix(n-1, T, decls_fin(Nil), m, ctx, s, z, U, Ds))
case tp_and(T1, T2) =>
n>0 &&
exists Ds1, Ds2 :: expansion_iter(n-1, m, ctx, s, z, T1, Ds1) &&
expansion_iter(n-1, m, ctx, s, z, T2, Ds2) &&
Ds==decls_and(Ds1, Ds2)
case tp_or(T1, T2) =>
n>0 &&
exists Ds1, Ds2 :: expansion_iter(n-1, m, ctx, s, z, T1, Ds1) &&
expansion_iter(n-1, m, ctx, s, z, T2, Ds2) &&
Ds==decls_or(Ds1, Ds2)
case tp_top => Ds==decls_fin(Nil)
case tp_bot => Ds==decls_bot
}
predicate expansion_fix(n: nat, selT: tp, selDs: decls, m: seq<pair<tp, decls>>, ctx: context, s: store, z: nat, T: tp, Ds: decls)
decreases n;
{
n>0 && (
(selDs==Ds && expansion_iter(n-1, [P(selT, selDs)]+m, ctx, s, z, T, Ds)) ||
(selDs!=Ds && exists Da :: expansion_iter(n-1, [P(selT, selDs)]+m, ctx, s, z, T, Da) &&
expansion_fix(n-1, selT, Da, m, ctx, s, z, T, Ds)))
}
predicate decl_sub(n: nat, ctx: context, s: store, d1: decl, d2: decl)
requires decl_eq(d1, d2);
decreases n;
{
match d1
case decl_tp(L, S, U, concrete) => n>0 && subtype(n-1, ctx, s, d2.S, S) && subtype(n-1, ctx, s, U, d2.U)
case decl_tm(l, U) => n>0 && subtype(n-1, ctx, s, U, d2.T)
case decl_mt(m, S, U) => n>0 && subtype(n-1, ctx, s, d2.P, S) && subtype(n-1, ctx, s, U, d2.R)
}
predicate decls_fin_sub(n: nat, ctx: context, s: store, s1: seq<decl>, s2: seq<decl>)
decreases n;
{
(s1 == [] && s2 == []) ||
(|s1|>0 && |s2|>0 && n>0 && (
(decl_eq(s1[0], s2[0]) && decl_sub(n-1, ctx, s, s1[0], s2[0]) &&
decls_fin_sub(n-1, ctx, s, s1[1..], s2[1..])) ||
(decl_lt(s1[0], s2[0]) && decls_fin_sub(n-1, ctx, s, s1[1..], s2))))
}
predicate decls_sub(n: nat, ctx: context, s: store, Ds1: decls, Ds2: decls)
decreases n;
{
match Ds1
case decls_bot => true
case decls_fin(s1) =>
(match Ds2
case decls_bot => false
case decls_fin(s2) => n>0 && decls_fin_sub(n-1, ctx, s, lst2seq(s1), lst2seq(s2)))
}
predicate path_red(ctx: context, s: store, p1: tm, p2: tm)
{
path(p1) && path(p2) && (
(p1.tm_sel? && p1.t.tm_loc? && p2.tm_loc? && p1.t.loc < |s.m| &&
def_field_lookup(p1.l, store_lookup(p1.t.loc, s)).Some? &&
p2==def_field_lookup(p1.l, store_lookup(p1.t.loc, s)).get) ||
(p1.tm_sel? && p2.tm_sel? && p1.l==p2.l && path_red(ctx, s, p1.t, p2.t)))
}
predicate subtype(n: nat, ctx: context, s: store, S: tp, T: tp)
decreases n;
{
var self := fresh_in_context(ctx);
n>0 && (
/* refl */ (S==T && wfe_type(n-1, ctx, s, T)) ||
/* <:-top */ (T.tp_top? && wfe_type(n-1, ctx, s, S)) ||
/* bot-<: */ (S.tp_bot? && wfe_type(n-1, ctx, s, T)) ||
/* <:-rfn */ (T.tp_rfn? && wfe_type(n-1, ctx, s, T) && subtype(n-1, ctx, s, S, T.base_tp) &&
exists Ds' :: expansion(n-1, ctx, s, self, S, Ds') &&
exists rfn_decls :: rfn_decls==decl_seq_sort(lst2seq(decls_subst(T.self, tm_var(self), T.decls))) &&
decls_sub(n-1, context_extend(ctx, self, S), s, decls_fin(seq2lst(rfn_decls)), Ds')) ||
/* rfn-<: */ (S.tp_rfn? && wfe_type(n-1, ctx, s, S) && subtype(n-1, ctx, s, S.base_tp, T)) ||
/* <:-sel */ (T.tp_sel? &&
exists S', U' :: type_membership(n-1, ctx, s, T.p, T.L, T.concrete, S', U') &&
subtype(n-1, ctx, s, S', U') && subtype(n-1, ctx, s, S, S')) ||
/* sel-<: */ (S.tp_sel? &&
exists S', U' :: type_membership(n-1, ctx, s, S.p, S.L, S.concrete, S', U') &&
subtype(n-1, ctx, s, S', U') && subtype(n-1, ctx, s, U', T)) ||
/* <:-and */ (T.tp_and? && subtype(n-1, ctx, s, S, T.and1) && subtype(n-1, ctx, s, S, T.and2)) ||
/* and1-<: */ (S.tp_and? && wfe_type(n-1, ctx, s, S.and2) && subtype(n-1, ctx, s, S.and1, T)) ||
/* and2-<: */ (S.tp_and? && wfe_type(n-1, ctx, s, S.and1) && subtype(n-1, ctx, s, S.and2, T)) ||
/* <:-or1 */ (T.tp_or? && wfe_type(n-1, ctx, s, T.or2) && subtype(n-1, ctx, s, S, T.or1)) ||
/* <:-or2 */ (T.tp_or? && wfe_type(n-1, ctx, s, T.or1) && subtype(n-1, ctx, s, S, T.or2)) ||
/* or-<: */ (S.tp_or? && subtype(n-1, ctx, s, S.or1, T) && subtype(n-1, ctx, s, S.or2, T)) ||
/* pathred */ (T.tp_sel? && wfe_type(n-1, ctx, s, T) && exists p :: path_red(ctx, s, T.p, p) &&
subtype(n-1, ctx, s, S, tp_sel(p, T.L, T.concrete))))
}
predicate typing'(ctx: context, s: store, t: tm, T: tp)
{
exists n:nat :: typing(n, ctx, s, t, T)
}
predicate wf_init'(already_in_store: bool, ctx: context, s: store, decls: seq<decl>, defs: seq<def>)
{
exists n:nat :: wf_init(n, already_in_store, ctx, s, decls, defs)
}
predicate wfe_type'(ctx: context, s: store, T: tp)
{
exists n:nat :: wfe_type(n, ctx, s, T)
}
predicate membership'(ctx: context, s: store, t: tm, l: nat, d: decl)
{
exists n:nat :: membership(n, ctx, s, t, l, d)
}
predicate field_membership'(ctx: context, s: store, t: tm, l: nat, T: tp)
{
exists n:nat :: field_membership(n, ctx, s, t, l, T)
}
predicate method_membership'(ctx: context, s: store, t: tm, m: nat, P: tp, R: tp)
{
exists n:nat :: method_membership(n, ctx, s, t, m, P, R)
}
predicate expansion'(ctx: context, s: store, z: nat, T: tp, Ds: decls)
{
exists n:nat :: expansion(n, ctx, s, z, T, Ds)
}
predicate decl_sub'(ctx: context, s: store, d1: decl, d2: decl)
requires decl_eq(d1, d2);
{
exists n:nat :: decl_sub(n, ctx, s, d1, d2)
}
predicate subtype'(ctx: context, s: store, S: tp, T: tp)
{
exists n:nat :: subtype(n, ctx, s, S, T)
}
// ### Properties about typing-related judgments ###
function typing_n(ctx: context, s: store, t: tm, T: tp): nat
requires typing'(ctx, s, t, T);
ensures typing(typing_n(ctx, s, t, T), ctx, s, t, T);
{
var n:nat :| typing(n, ctx, s, t, T); n
}
function wf_init_n(already_in_store: bool, ctx: context, s: store, decls: seq<decl>, defs: seq<def>): nat
requires wf_init'(already_in_store, ctx, s, decls, defs);
ensures wf_init(wf_init_n(already_in_store, ctx, s, decls, defs), already_in_store, ctx, s, decls, defs);
{
var n:nat :| wf_init(n, already_in_store, ctx, s, decls, defs); n
}
function wfe_type_n(ctx: context, s: store, T: tp): nat
requires wfe_type'(ctx, s, T);
ensures wfe_type(wfe_type_n(ctx, s, T), ctx, s, T);
{
var n:nat :| wfe_type(n, ctx, s, T); n
}
function field_membership_n(ctx: context, s: store, t: tm, l: nat, T: tp): nat
requires field_membership'(ctx, s, t, l, T);
ensures field_membership(field_membership_n(ctx, s, t, l, T), ctx, s, t, l, T);
{
var n:nat :| field_membership(n, ctx, s, t, l, T); n
}
function method_membership_n(ctx: context, s: store, t: tm, m: nat, P: tp, R: tp): nat
requires method_membership'(ctx, s, t, m, P, R);
ensures method_membership(method_membership_n(ctx, s, t, m, P, R), ctx, s, t, m, P, R);
{
var n:nat :| method_membership(n, ctx, s, t, m, P, R); n
}
function expansion_n(ctx: context, s: store, z: nat, T: tp, Ds: decls): nat
requires expansion'(ctx, s, z, T, Ds);
ensures expansion(expansion_n(ctx, s, z, T, Ds), ctx, s, z, T, Ds);
{
var n:nat :| expansion(n, ctx, s, z, T, Ds); n
}
function subtype_n(ctx: context, s: store, S: tp, T: tp): nat
requires subtype'(ctx, s, S, T);
ensures subtype(subtype_n(ctx, s, S, T), ctx, s, S, T);
{
var n:nat :| subtype(n, ctx, s, S, T); n
}
ghost method lemma_typing_monotonic_plus(m: nat, n: nat, ctx: context, s: store, t: tm, T: tp)
requires m<=n;
requires typing(m, ctx, s, t, T);
ensures typing(n, ctx, s, t, T);
decreases n-m;
{
if (n==m) {}
else {
lemma_typing_monotonic(m, ctx, s, t, T);
lemma_typing_monotonic_plus(m+1, n, ctx, s, t, T);
}
}