(** Adding or removing an equation from a set. *)
Program Definition add_equation (q: equation) (e: eqs) :=
mkeqs (EqSet.add q (eqs1 e)) (EqSet2.add q (eqs2 e)) _.
Next Obligation.
split; intros.
destruct (OrderedEquation'.eq_dec q q0).
apply EqSet.add_1; auto.
apply EqSet.add_2. apply (eqs_same e). apply EqSet2.add_3 with q; auto.
destruct (OrderedEquation.eq_dec q q0).
apply EqSet2.add_1; auto.
apply EqSet2.add_2. apply (eqs_same e). apply EqSet.add_3 with q; auto.
Qed.
Program Definition remove_equation (q: equation) (e: eqs) :=
mkeqs (EqSet.remove q (eqs1 e)) (EqSet2.remove q (eqs2 e)) _.
Next Obligation.
split; intros.
destruct (OrderedEquation'.eq_dec q q0).
eelim EqSet2.remove_1; eauto.
apply EqSet.remove_2; auto. apply (eqs_same e). apply EqSet2.remove_3 with q; auto.
destruct (OrderedEquation.eq_dec q q0).
eelim EqSet.remove_1; eauto.
apply EqSet2.remove_2; auto. apply (eqs_same e). apply EqSet.remove_3 with q; auto.
Qed.
(** [reg_unconstrained r e] is true if [e] contains no equations involving the RTL pseudoregister [r].  In other words, all equations [r' = l [kind]] in [e] are such that [r' <> r]. *)
Definition select_reg_l (r: reg) (q: equation) := Pos.leb r (ereg q).
Definition select_reg_h (r: reg) (q: equation) := Pos.leb (ereg q) r.
Definition reg_unconstrained (r: reg) (e: eqs) : bool :=
negb (EqSet.mem_between (select_reg_l r) (select_reg_h r) (eqs1 e)).
(** [loc_unconstrained l e] is true if [e] contains no equations involving the LTL location [l] or a location that partially overlaps with [l]. In other words, all equations [r = l' [kind]] in [e] are such that [Loc.diff l' l]. *)
Definition select_loc_l (l: loc) :=
let lb := OrderedLoc.diff_low_bound l in
fun (q: equation) => match OrderedLoc.compare (eloc q) lb with LT _ => false | _ => true end.
Definition select_loc_h (l: loc) :=
let lh := OrderedLoc.diff_high_bound l in
fun (q: equation) => match OrderedLoc.compare (eloc q) lh with GT _ => false | _ => true end.
Definition loc_unconstrained (l: loc) (e: eqs) : bool :=
negb (EqSet2.mem_between (select_loc_l l) (select_loc_h l) (eqs2 e)).
Definition reg_loc_unconstrained (r: reg) (l: loc) (e: eqs) : bool :=
reg_unconstrained r e && loc_unconstrained l e.
(** [subst_reg r1 r2 e] simulates the effect of assigning [r2] to [r1] on [e]. All equations of the form [r1 = l [kind]] are replaced by [r2 = l [kind]]. *)
Definition subst_reg (r1 r2: reg) (e: eqs) : eqs :=
EqSet.fold
(fun q e => add_equation (Eq (ekind q) r2 (eloc q)) (remove_equation q e))
(EqSet.elements_between (select_reg_l r1) (select_reg_h r1) (eqs1 e))
e.
(** [subst_reg_kind r1 k1 r2 k2 e] simulates the effect of assigning the [k2] part of [r2] to the [k1] part of [r1] on [e]. All equations of the form [r1 = l [k1]] are replaced by [r2 = l [k2]]. *)
Definition subst_reg_kind (r1: reg) (k1: equation_kind) (r2: reg) (k2: equation_kind) (e: eqs) : eqs :=
EqSet.fold
(fun q e =>
if IndexedEqKind.eq (ekind q) k1
then add_equation (Eq k2 r2 (eloc q)) (remove_equation q e)
else e)
(EqSet.elements_between (select_reg_l r1) (select_reg_h r1) (eqs1 e))
e.
(** [subst_loc l1 l2 e] simulates the effect of assigning [l2] to [l1] on [e]. All equations of the form [r = l1 [kind]] are replaced by [r = l2 [kind]]. Return [None] if [e] contains an equation of the form [r = l] with [l] partially overlapping [l1]. *)
Definition subst_loc (l1 l2: loc) (e: eqs) : option eqs :=
EqSet2.fold
(fun q opte =>
match opte with
| None => None
| Some e =>
if Loc.eq l1 (eloc q) then
Some (add_equation (Eq (ekind q) (ereg q) l2) (remove_equation q e))
else
None
end)
(EqSet2.elements_between (select_loc_l l1) (select_loc_h l1) (eqs2 e))
(Some e).
(** [subst_loc_part l1 l2 k e] simulates the effect of assigning [l2] to the [k] part of [l1] on [e]. All equations of the form [r = l1 [k]] are replaced by [r = l2 [Full]]. Return [None] if [e] contains an equation of the form [r = l] with [l] partially overlapping [l1], or an equation of the form [r = l1] with a kind different from [k1]. *)
Definition subst_loc_part (l1: loc) (l2: loc) (k: equation_kind) (e: eqs) : option eqs :=
EqSet2.fold
(fun q opte =>
match opte with
| None => None
| Some e =>
if Loc.eq l1 (eloc q) then
if IndexedEqKind.eq (ekind q) k
then Some (add_equation (Eq Full (ereg q) l2) (remove_equation q e))
else None
else
None
end)
(EqSet2.elements_between (select_loc_l l1) (select_loc_h l1) (eqs2 e))
(Some e).
(** [subst_loc_pair l1 l2 l2'] simulates the effect of assigning [makelong l2 l2'] to [l1].  All equations of the form [r = l1 [Full]] are replaced by the two equations [r = l2 [High], r = l2' [Low]]. Return [None] if [e] contains an equation of the form [r = l] with [l] partially overlapping [l1], or an equation of the form [r = l1] with a kind different from [Full]. *)
Definition subst_loc_pair (l1 l2 l2': loc) (e: eqs) : option eqs :=
EqSet2.fold
(fun q opte =>
match opte with
| None => None
| Some e =>
if Loc.eq l1 (eloc q) then
if IndexedEqKind.eq (ekind q) Full
then Some (add_equation (Eq High (ereg q) l2)
(add_equation (Eq Low (ereg q) l2')
(remove_equation q e)))
else None
else
None
end)
(EqSet2.elements_between (select_loc_l l1) (select_loc_h l1) (eqs2 e))
(Some e).
(** [loc_type_compat env l e] checks that for all equations [r = l] in [e], the type [env r] of [r] is compatible with the type of [l]. *)
Definition sel_type (k: equation_kind) (ty: typ) : typ :=
match k with
| Full => ty
| Low | High => Tint
end.
Definition loc_type_compat (env: regenv) (l: loc) (e: eqs) : bool :=
EqSet2.for_all_between
(fun q => subtype (sel_type (ekind q) (env (ereg q))) (Loc.type l))
(select_loc_l l) (select_loc_h l) (eqs2 e).
(** [long_type_compat env l e] checks that for all equations [r = l] in [e]. then type [env r] of [r] is compatible with the type [Tlong]. *)
Definition long_type_compat (env: regenv) (l: loc) (e: eqs) : bool :=
EqSet2.for_all_between
(fun q => subtype (env (ereg q)) Tlong)
(select_loc_l l) (select_loc_h l) (eqs2 e).
(** [add_equations [r1...rN] [m1...mN] e] adds to [e] the [N] equations [ri = R mi [Full]].  Return [None] if the two lists have different lengths. *)
Fixpoint add_equations (rl: list reg) (ml: list mreg) (e: eqs) : option eqs :=
match rl, ml with
| nil, nil => Some e
| r1 :: rl, m1 :: ml => add_equations rl ml (add_equation (Eq Full r1 (R m1)) e)
| _, _ => None
end.
(** [add_equations_args] is similar but additionally handles the splitting of pseudoregisters of type [Tlong] in two locations containing the two 32-bit halves of the 64-bit integer. *)
Function add_equations_args (rl: list reg) (tyl: list typ) (ll: list (rpair loc)) (e: eqs) : option eqs :=
match rl, tyl, ll with
| nil, nil, nil => Some e
| r1 :: rl, ty :: tyl, One l1 :: ll =>
add_equations_args rl tyl ll (add_equation (Eq Full r1 l1) e)
| r1 :: rl, Tlong :: tyl, Twolong l1 l2 :: ll =>
if Archi.ptr64 then None else
add_equations_args rl tyl ll (add_equation (Eq Low r1 l2) (add_equation (Eq High r1 l1) e))
| _, _, _ => None
end.
(** [add_equations_res] is similar but is specialized to the case where there is only one pseudo-register. *)
Function add_equations_res (r: reg) (oty: option typ) (p: rpair mreg) (e: eqs) : option eqs :=
match p, oty with
| One mr, _ =>
Some (add_equation (Eq Full r (R mr)) e)
| Twolong mr1 mr2, Some Tlong =>
if Archi.ptr64 then None else
Some (add_equation (Eq Low r (R mr2)) (add_equation (Eq High r (R mr1)) e))
| _, _ =>
None
end.
(** [remove_equations_res] is similar to [add_equations_res] but removes equations instead of adding them. *)
Function remove_equations_res (r: reg) (p: rpair mreg) (e: eqs) : option eqs :=
match p with
| One mr =>
Some (remove_equation (Eq Full r (R mr)) e)
| Twolong mr1 mr2 =>
if mreg_eq mr2 mr1
then None
else Some (remove_equation (Eq Low r (R mr2)) (remove_equation (Eq High r (R mr1)) e))
end.
(** [add_equations_ros] adds an equation, if needed, between an optional pseudoregister and an optional machine register.  It is used for the function argument of the [Icall] and [Itailcall] instructions. *)
Definition add_equation_ros (ros: reg + ident) (ros': mreg + ident) (e: eqs) : option eqs :=
match ros, ros' with
| inl r, inl mr => Some(add_equation (Eq Full r (R mr)) e)
| inr id, inr id' => assertion (ident_eq id id'); Some e
| _, _ => None
end.
(** [add_equations_builtin_arg] adds the needed equations for arguments to builtin functions. *)
Fixpoint add_equations_builtin_arg
(env: regenv) (arg: builtin_arg reg) (arg': builtin_arg loc) (e: eqs) : option eqs :=
match arg, arg' with
| BA r, BA l =>
Some (add_equation (Eq Full r l) e)
| BA r, BA_splitlong (BA lhi) (BA llo) =>
assertion (typ_eq (env r) Tlong);
assertion (Archi.splitlong);
Some (add_equation (Eq Low r llo) (add_equation (Eq High r lhi) e))
| BA_int n, BA_int n' =>
assertion (Int.eq_dec n n'); Some e
| BA_long n, BA_long n' =>
assertion (Int64.eq_dec n n'); Some e
| BA_float f, BA_float f' =>
assertion (Float.eq_dec f f'); Some e
| BA_single f, BA_single f' =>
assertion (Float32.eq_dec f f'); Some e
| BA_loadstack chunk ofs, BA_loadstack chunk' ofs' =>
assertion (chunk_eq chunk chunk');
assertion (Ptrofs.eq_dec ofs ofs');
Some e
| BA_addrstack ofs, BA_addrstack ofs' =>
assertion (Ptrofs.eq_dec ofs ofs');
Some e
| BA_loadglobal chunk id ofs, BA_loadglobal chunk' id' ofs' =>
assertion (chunk_eq chunk chunk');
assertion (ident_eq id id');
assertion (Ptrofs.eq_dec ofs ofs');
Some e
| BA_addrglobal id ofs, BA_addrglobal id' ofs' =>
assertion (ident_eq id id');
assertion (Ptrofs.eq_dec ofs ofs');
Some e
| BA_splitlong hi lo, BA_splitlong hi' lo' =>
do e1 <- add_equations_builtin_arg env hi hi' e;
add_equations_builtin_arg env lo lo' e1
| BA_addptr a1 a2, BA_addptr a1' a2' =>
do e1 <- add_equations_builtin_arg env a1 a1' e;
add_equations_builtin_arg env a2 a2' e1
| _, _ =>
None
end.
Fixpoint add_equations_builtin_args
(env: regenv) (args: list (builtin_arg reg))
(args': list (builtin_arg loc)) (e: eqs) : option eqs :=
match args, args' with
| nil, nil => Some e
| a1 :: al, a1' :: al' =>
do e1 <- add_equations_builtin_arg env a1 a1' e;
add_equations_builtin_args env al al' e1
| _, _ => None
end.
(** For [EF_debug] builtins, some arguments can be removed. *)
Fixpoint add_equations_debug_args
(env: regenv) (args: list (builtin_arg reg))
(args': list (builtin_arg loc)) (e: eqs) : option eqs :=
match args, args' with
| _, nil => Some e
| a1 :: al, a1' :: al' =>
match add_equations_builtin_arg env a1 a1' e with
| None => add_equations_debug_args env al args' e
| Some e1 => add_equations_debug_args env al al' e1
end
| _, _ => None
end.
(** Checking of the result of a builtin *)
Definition remove_equations_builtin_res
(env: regenv) (res: builtin_res reg) (res': builtin_res mreg) (e: eqs) : option eqs :=
match res, res' with
| BR r, BR r' => Some (remove_equation (Eq Full r (R r')) e)
| BR r, BR_splitlong (BR rhi) (BR rlo) =>
assertion (typ_eq (env r) Tlong);
if mreg_eq rhi rlo then None else
Some (remove_equation (Eq Low r (R rlo))
(remove_equation (Eq High r (R rhi)) e))
| BR_none, BR_none => Some e
| _, _ => None
end.
(** [can_undef ml] returns true if all machine registers in [ml] are unconstrained and can harmlessly be undefined. *)
Fixpoint can_undef (ml: list mreg) (e: eqs) : bool :=
match ml with
| nil => true
| m1 :: ml => loc_unconstrained (R m1) e && can_undef ml e
end.
Fixpoint can_undef_except (l: loc) (ml: list mreg) (e: eqs) : bool :=
match ml with
| nil => true
| m1 :: ml =>
(Loc.eq l (R m1) || loc_unconstrained (R m1) e) && can_undef_except l ml e
end.
(** [no_caller_saves e] returns [e] if all caller-save locations are unconstrained in [e].  In other words, [e] contains no equations involving a caller-save register or [Outgoing] stack slot. *)
Definition no_caller_saves (e: eqs) : bool :=
EqSet.for_all
(fun eq =>
match eloc eq with
| R r => is_callee_save r
| S Outgoing _ _ => false
| S _ _ _ => true
end)
e.
(** [compat_left r l e] returns true if all equations in [e] that involve [r] are of the form [r = l [Full]]. *)
Definition compat_left (r: reg) (l: loc) (e: eqs) : bool :=
EqSet.for_all_between
(fun q =>
match ekind q with
| Full => Loc.eq l (eloc q)
| _ => false
end)
(select_reg_l r) (select_reg_h r)
(eqs1 e).
(** [compat_left2 r l1 l2 e] returns true if all equations in [e] that involve [r] are of the form [r = l1 [High]] or [r = l2 [Low]]. *)
Definition compat_left2 (r: reg) (l1 l2: loc) (e: eqs) : bool :=
EqSet.for_all_between
(fun q =>
match ekind q with
| High => Loc.eq l1 (eloc q)
| Low => Loc.eq l2 (eloc q)
| _ => false
end)
(select_reg_l r) (select_reg_h r)
(eqs1 e).
(** [ros_compatible_tailcall ros] returns true if [ros] is a function name or a caller-save register.  This is used to check [Itailcall] instructions. *)
Definition ros_compatible_tailcall (ros: mreg + ident) : bool :=
match ros with
| inl r => negb (is_callee_save r)
| inr id => true
end.
(** * The validator *)
Definition destroyed_by_move (src dst: loc) :=
match src, dst with
| S sl ofs ty, _ => destroyed_by_getstack sl
| _, S sl ofs ty => destroyed_by_setstack ty
| _, _ => destroyed_by_op Omove
end.
Definition well_typed_move (env: regenv) (dst: loc) (e: eqs) : bool :=
match dst with
| R r => true
| S sl ofs ty => loc_type_compat env dst e
end.
(** Simulate the effect of a sequence of moves [mv] on a set of equations [e].  The set [e] is the equations that must hold after the sequence of moves.  Return the set of equations that must hold before the sequence of moves.  Return [None] if the set of equations [e] cannot hold after the sequence of moves. *)
Fixpoint track_moves (env: regenv) (mv: moves) (e: eqs) : option eqs :=
match mv with
| nil => Some e
| MV src dst :: mv =>
do e1 <- track_moves env mv e;
assertion (can_undef_except dst (destroyed_by_move src dst)) e1;
assertion (well_typed_move env dst e1);
subst_loc dst src e1
| MVmakelong src1 src2 dst :: mv =>
assertion (negb Archi.ptr64);
do e1 <- track_moves env mv e;
assertion (long_type_compat env (R dst) e1);
subst_loc_pair (R dst) (R src1) (R src2) e1
| MVlowlong src dst :: mv =>
assertion (negb Archi.ptr64);
do e1 <- track_moves env mv e;
subst_loc_part (R dst) (R src) Low e1
| MVhighlong src dst :: mv =>
assertion (negb Archi.ptr64);
do e1 <- track_moves env mv e;
subst_loc_part (R dst) (R src) High e1
end.
(** [transfer_use_def args res args' res' undefs e] returns the set of equations that must hold "before" in order for the equations [e] to hold "after" the execution of RTL and LTL code of the following form: << RTL                            LTL use pseudoregs args            use machine registers args' define pseudoreg res           undefine machine registers undef define machine register res' >> As usual, [None] is returned if the equations [e] cannot hold after this execution. *)
Definition transfer_use_def (args: list reg) (res: reg) (args': list mreg) (res': mreg)
(undefs: list mreg) (e: eqs) : option eqs :=
let e1 := remove_equation (Eq Full res (R res')) e in
assertion (reg_loc_unconstrained res (R res') e1);
assertion (can_undef undefs e1);
add_equations args args' e1.
Definition kind_first_word := if Archi.big_endian then High else Low.
Definition kind_second_word := if Archi.big_endian then Low else High.
(** The core transfer function.  It takes a set [e] of equations that must hold "after" and a block shape [shape] representing a matching pair of an RTL instruction and an LTL basic block.  It returns the set of equations that must hold "before" these instructions, or [None] if impossible. *)
Definition transfer_aux (f: RTL.function) (env: regenv)
(shape: block_shape) (e: eqs) : option eqs :=
match shape with
| BSnop mv s =>
track_moves env mv e
| BSmove src dst mv s =>
track_moves env mv (subst_reg dst src e)
| BSmakelong src1 src2 dst mv s =>
let e1 := subst_reg_kind dst High src1 Full e in
let e2 := subst_reg_kind dst Low src2 Full e1 in
assertion (reg_unconstrained dst e2);
track_moves env mv e2
| BSlowlong src dst mv s =>
let e1 := subst_reg_kind dst Full src Low e in
assertion (reg_unconstrained dst e1);
track_moves env mv e1
| BShighlong src dst mv s =>
let e1 := subst_reg_kind dst Full src High e in
assertion (reg_unconstrained dst e1);
track_moves env mv e1
| BSop op args res mv1 args' res' mv2 s =>
do e1 <- track_moves env mv2 e;
do e2 <- transfer_use_def args res args' res' (destroyed_by_op op) e1;
track_moves env mv1 e2
| BSopdead op args res mv s =>
assertion (reg_unconstrained res e);
track_moves env mv e
| BSload chunk addr args dst mv1 args' dst' mv2 s =>
do e1 <- track_moves env mv2 e;
do e2 <- transfer_use_def args dst args' dst' (destroyed_by_load chunk addr) e1;
track_moves env mv1 e2
| BSload2 addr addr' args dst mv1 args1' dst1' mv2 args2' dst2' mv3 s =>
do e1 <- track_moves env mv3 e;
let e2 := remove_equation (Eq kind_second_word dst (R dst2')) e1 in
assertion (loc_unconstrained (R dst2') e2);
assertion (can_undef (destroyed_by_load Mint32 addr') e2);
do e3 <- add_equations args args2' e2;
do e4 <- track_moves env mv2 e3;
let e5 := remove_equation (Eq kind_first_word dst (R dst1')) e4 in
assertion (loc_unconstrained (R dst1') e5);
assertion (can_undef (destroyed_by_load Mint32 addr) e5);
assertion (reg_unconstrained dst e5);
do e6 <- add_equations args args1' e5;
track_moves env mv1 e6
| BSload2_1 addr args dst mv1 args' dst' mv2 s =>
do e1 <- track_moves env mv2 e;
let e2 := remove_equation (Eq kind_first_word dst (R dst')) e1 in
assertion (reg_loc_unconstrained dst (R dst') e2);
assertion (can_undef (destroyed_by_load Mint32 addr) e2);
do e3 <- add_equations args args' e2;
track_moves env mv1 e3
| BSload2_2 addr addr' args dst mv1 args' dst' mv2 s =>
do e1 <- track_moves env mv2 e;
let e2 := remove_equation (Eq kind_second_word dst (R dst')) e1 in
assertion (reg_loc_unconstrained dst (R dst') e2);
assertion (can_undef (destroyed_by_load Mint32 addr') e2);
do e3 <- add_equations args args' e2;
track_moves env mv1 e3
| BSloaddead chunk addr args dst mv s =>
assertion (reg_unconstrained dst e);
track_moves env mv e
| BSstore chunk addr args src mv args' src' s =>
assertion (can_undef (destroyed_by_store chunk addr) e);
do e1 <- add_equations (src :: args) (src' :: args') e;
track_moves env mv e1
| BSstore2 addr addr' args src mv1 args1' src1' mv2 args2' src2' s =>
assertion (can_undef (destroyed_by_store Mint32 addr') e);
do e1 <- add_equations args args2'
(add_equation (Eq kind_second_word src (R src2')) e);
do e2 <- track_moves env mv2 e1;
assertion (can_undef (destroyed_by_store Mint32 addr) e2);
do e3 <- add_equations args args1'
(add_equation (Eq kind_first_word src (R src1')) e2);
track_moves env mv1 e3
| BScall sg ros args res mv1 ros' mv2 s =>
let args' := loc_arguments sg in
let res' := loc_result sg in
do e1 <- track_moves env mv2 e;
do e2 <- remove_equations_res res res' e1;
assertion (forallb (fun l => reg_loc_unconstrained res l e2)
(map R (regs_of_rpair res')));
assertion (no_caller_saves e2);
do e3 <- add_equation_ros ros ros' e2;
do e4 <- add_equations_args args (sig_args sg) args' e3;
track_moves env mv1 e4
| BStailcall sg ros args mv1 ros' =>
let args' := loc_arguments sg in
assertion (tailcall_is_possible sg);
assertion (opt_typ_eq sg.(sig_res) f.(RTL.fn_sig).(sig_res));
assertion (ros_compatible_tailcall ros');
do e1 <- add_equation_ros ros ros' empty_eqs;
do e2 <- add_equations_args args (sig_args sg) args' e1;
track_moves env mv1 e2
| BSbuiltin ef args res mv1 args' res' mv2 s =>
do e1 <- track_moves env mv2 e;
do e2 <- remove_equations_builtin_res env res res' e1;
assertion (forallb (fun r => reg_unconstrained r e2)
(params_of_builtin_res res));
assertion (forallb (fun mr => loc_unconstrained (R mr) e2)
(params_of_builtin_res res'));
assertion (can_undef (destroyed_by_builtin ef) e2);
do e3 <-
match ef with
| EF_debug _ _ _ => add_equations_debug_args env args args' e2
| _              => add_equations_builtin_args env args args' e2
end;
track_moves env mv1 e3
| BScond cond args mv args' s1 s2 =>
assertion (can_undef (destroyed_by_cond cond) e);
do e1 <- add_equations args args' e;
track_moves env mv e1
| BSjumptable arg mv arg' tbl =>
assertion (can_undef destroyed_by_jumptable e);
track_moves env mv (add_equation (Eq Full arg (R arg')) e)
| BSreturn None mv =>
track_moves env mv empty_eqs
| BSreturn (Some arg) mv =>
let arg' := loc_result (RTL.fn_sig f) in
do e1 <- add_equations_res arg (sig_res (RTL.fn_sig f)) arg' empty_eqs;
track_moves env mv e1
end.
(** The main transfer function for the dataflow analysis.  Like [transfer_aux], it infers the equations that must hold "before" as a function of the equations that must hold "after".  It also handles error propagation and reporting. *)
Definition transfer (f: RTL.function) (env: regenv) (shapes: PTree.t block_shape)
(pc: node) (after: res eqs) : res eqs :=
match after with
| Error _ => after
| OK e =>
match shapes!pc with
| None => Error(MSG "At PC " :: POS pc :: MSG ": unmatched block" :: nil)
| Some shape =>
match transfer_aux f env shape e with
| None => Error(MSG "At PC " :: POS pc :: MSG ": invalid register allocation" :: nil)
| Some e' => OK e'
end
end
end.
(** The semilattice for dataflow analysis.  Operates on analysis results of type [res eqs], that is, either a set of equations or an error message.  Errors correspond to [Top].  Sets of equations are ordered by inclusion. *)
Module LEq <: SEMILATTICE.
Definition t := res eqs.
Definition eq (x y: t) :=
match x, y with
| OK a, OK b => EqSet.Equal a b
| Error _, Error _ => True
| _, _ => False
end.

Lemma ge_refl: forall x y, eq x y -> ge x y.