Source file HO_unif.ml
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(** {1 Higher-Order Unification} *)
module RW = Rewrite
module T = Term
module US = Unif_subst
let stat_unif_calls = Util.mk_stat "ho_unif.calls"
let stat_unif_steps = Util.mk_stat "ho_unif.steps"
let prof_norm_subst = Util.mk_profiler "ho_unif.norm_subst"
let section = Util.Section.make "ho_unif"
type term = Term.t
type penalty = int
(** penalty on the search space *)
type pair = Type.t list * term * term
let default_fuel = ref 15
let enable_norm_subst = ref true
let term_arity args =
args
|> Util.take_drop_while (fun t -> T.is_type t)
|> CCPair.map List.length List.length
let enum_prop ?(mode=`Full) ((v:Term.var), sc_v) ~offset : (Subst.t * penalty) list =
let ty_v = HVar.ty v in
let n, ty_args, ty_ret = Type.open_poly_fun ty_v in
assert (Type.is_prop ty_ret);
if n>0 then []
else (
let vars = List.mapi (fun i ty -> HVar.make ~ty i) ty_args in
let l_not = match mode with
| `None -> None
| `Neg | `Full ->
let f = HVar.make offset ~ty:ty_v in
T.fun_of_fvars vars
(T.Form.not_ (T.app (T.var f) (List.map T.var vars)))
|> CCOpt.return
and l_and = match mode with
| `Neg | `None -> None
| `Full ->
let f = HVar.make offset ~ty:ty_v in
let g = HVar.make (offset+1) ~ty:ty_v in
T.fun_of_fvars vars
(T.Form.and_
(T.app (T.var f) (List.map T.var vars))
(T.app (T.var g) (List.map T.var vars)))
|> CCOpt.return
and l_eq = match mode with
| `Neg | `None -> None
| `Full ->
let a = HVar.make offset ~ty:Type.tType in
let ty_fun = Type.arrow ty_args (Type.var a) in
let f = HVar.make (offset+1) ~ty:ty_fun in
let g = HVar.make (offset+2) ~ty:ty_fun in
T.fun_of_fvars vars
(T.Form.eq
(T.app (T.var f) (List.map T.var vars))
(T.app (T.var g) (List.map T.var vars)))
|> CCOpt.return
in
CCList.filter_map
(fun (o,penalty) -> match o with
| None -> None
| Some t ->
assert (T.DB.is_closed t);
let subst = Subst.FO.bind' Subst.empty (v,sc_v) (t,sc_v) in
Some (subst, penalty))
[ l_not, 2;
l_and, 5;
l_eq, 10;
]
)
let pp_pair out ((env,t,u):pair) =
Format.fprintf out "(@[@[`%a` =?=@]@ `%a`@ :env [@[%a@]]@])"
T.pp t T.pp u (Util.pp_list ~sep:", " Type.pp) env
module U = struct
type pb = {
pairs: pair list;
subst: US.t;
penalty: penalty;
offset: int;
}
type state = {
sc: Scoped.scope;
mutable fuel: int;
queue: pb Queue.t;
offset0: int;
mutable sols: (US.t * penalty) list;
}
let empty sc fuel offset =
{ sc; fuel; queue=Queue.create(); sols=[]; offset0=offset; }
let add (st:state) pb : unit = Queue.push pb st.queue
let pp_pb out (pb:pb) =
Format.fprintf out "(@[pb :subst %a@ :pairs (@[<hv>%a@])@])"
US.pp pb.subst (Util.pp_list ~sep:" " pp_pair) pb.pairs
let pp out (t:state): unit =
Format.fprintf out
"(@[<hv2>ho_unif_pb@ %a@])"
(Util.pp_seq ~sep:" " pp_pb) (Iter.of_queue t.queue)
(** {6 normalization of pairs} *)
type pair_kind =
| P_rigid_rigid
| P_flex_rigid
| P_flex_flex
let classify_pair (_,t,u) = match T.is_ho_app t, T.is_ho_app u with
| false, false -> P_rigid_rigid
| false, true
| true, false -> P_flex_rigid
| true, true -> P_flex_flex
let is_flex_flex p = classify_pair p = P_flex_flex
let cmp_pairs p1 p2 : int =
let hardness =
function P_rigid_rigid -> 0 | P_flex_rigid -> 1 | P_flex_flex -> 2
in
CCOrd.int (classify_pair p1 |> hardness) (classify_pair p2 |> hardness)
let whnf_deref (subst:US.t) (t,sc) =
let t = match T.view t with
| T.Var _ -> US.FO.deref subst (t,sc) |> fst
| T.App (f, l) when T.is_var f ->
T.app (US.FO.deref subst (f,sc) |> fst) l
| _ -> t
in
Lambda.whnf t
let mk_pairs env l1 l2 = List.map2 (fun t u -> env, t, u) l1 l2
let flatten_rigid_rigid sc subst pairs : pair list option =
try
let rec aux acc l = match l with
| [] -> acc
| (env,t, u) :: tail ->
let t = whnf_deref subst (t,sc) in
let u = whnf_deref subst (u,sc) in
begin match T.Classic.view t, T.Classic.view u with
| _ when T.equal t u -> aux acc tail
| T.Classic.App (id1, l1), T.Classic.App (id2, l2) ->
if ID.equal id1 id2 && List.length l1 = List.length l2
then aux acc (mk_pairs env l1 l2 @ tail)
else raise Exit
| T.Classic.AppBuiltin (b1,l1), T.Classic.AppBuiltin (b2,l2) ->
if Builtin.equal b1 b2 && List.length l1=List.length l2
then aux acc (mk_pairs env l1 l2 @ tail)
else raise Exit
| _ -> aux ((env,t,u) :: acc) tail
end
in
Some (aux [] pairs)
with Exit ->
None
let mk_pb ~subst ~penalty ~offset pairs : pb =
{ subst; penalty; offset; pairs; }
let normalize_pb sc (pb:pb): pb option =
begin match flatten_rigid_rigid sc pb.subst pb.pairs with
| None -> None
| Some pairs ->
let pairs = List.sort cmp_pairs pairs in
Some { pb with pairs; }
end
(** {6 Main loop} *)
let mk_fresh_var offset ty = offset+1, HVar.make offset ~ty
let mk_fresh_vars offset ty_l = CCList.fold_map mk_fresh_var offset ty_l
let unif_lambda ~offset env v args t : pair list * int =
assert (T.is_fun t);
let ty_t_args, t_body = T.open_fun t in
assert (ty_t_args<>[]);
assert (not (T.is_fun t_body));
let all_args = List.map T.ty args @ ty_t_args in
let offset, v' =
let ty_v' = Type.arrow all_args (T.ty t_body) in
mk_fresh_var offset ty_v'
in
let bind_v =
let n = List.length all_args in
let rhs =
T.app (T.var v')
(List.mapi (fun i ty -> T.bvar ~ty (n-i-1)) all_args)
|> T.fun_l all_args
in
env, T.var v, rhs
and new_pair_body =
let n = List.length ty_t_args in
let lhs =
T.app (T.var v')
(List.map (T.DB.shift n) args @
List.mapi (fun i ty -> T.bvar ~ty (n-i-1)) ty_t_args) in
ty_t_args @ env, lhs, t_body
in
let new_pairs = [ bind_v; new_pair_body ] in
new_pairs, offset
let delay_pair (p:pair) sc : Unif_constr.t =
let env, t1, t2 = p in
let t1 = T.fun_l env t1 in
let t2 = T.fun_l env t2 in
let tags = [Proof.Tag.T_ho] in
Unif_constr.FO.make ~tags (t1,sc)(t2,sc)
let unif_rigid ~sc ~subst ~offset env v args t : (pair list * _ * _ * _) Iter.t =
assert (args<>[]);
let n_params, ty_args, ty_ret = Type.open_poly_fun (T.ty t) in
assert (n_params=0);
let n_ty_args = List.length ty_args in
let all_ty_args = List.map T.ty args @ ty_args in
let hd_t = T.head_term t in
let vars_right =
List.mapi (fun i ty -> T.bvar ~ty (n_ty_args-i-1)) ty_args
and vars_left =
let n = n_ty_args + List.length args in
List.mapi (fun i arg -> T.bvar ~ty:(T.ty arg) (n-i-1)) args
in
let all_vars = vars_left @ vars_right in
let rhs = T.app (T.DB.shift n_ty_args t) vars_right in
let lhs_args = List.map (T.DB.shift n_ty_args) args @ vars_right in
let proj =
Iter.of_list all_ty_args |> Util.seq_zipi
|> Iter.filter_map
(fun (i,ty_arg_i) ->
let ty_args_i, ty_ret_i = Type.open_fun ty_arg_i in
try
let subst = Unif.Ty.unify_full ~subst (ty_ret_i,sc) (ty_ret,sc) in
let offset, f_vars =
ty_args_i
|> List.map (Type.arrow all_ty_args)
|> mk_fresh_vars offset
in
let lambda =
let f_vars_applied =
List.map (fun f_var -> T.app (T.var f_var) all_vars) f_vars
in
T.app (List.nth all_vars i) f_vars_applied
|> T.fun_l (List.map T.ty all_vars)
and lhs =
let f_vars_applied =
List.map (fun f_var -> T.app (T.var f_var) lhs_args) f_vars
in
T.app (List.nth lhs_args i) f_vars_applied
in
let subst = US.FO.bind subst (v,sc) (lambda,sc) in
Some ([ty_args@env,lhs,rhs],subst,offset,"proj")
with Unif.Fail ->
None)
and imitate = match T.view hd_t with
| T.AppBuiltin (b,l) when vars_right=[] && T.args t=[] ->
let ty_params, l = CCList.take_drop_while T.is_type l in
let ty_args_t = List.map T.ty l in
let offset, f_vars =
ty_args_t
|> List.map (Type.arrow all_ty_args)
|> mk_fresh_vars offset
in
let lambda =
let f_vars_applied =
List.map (fun f_var -> T.app (T.var f_var) all_vars) f_vars
in
T.app_builtin ~ty:(T.ty t) b f_vars_applied
|> T.fun_l (List.map T.ty all_vars)
and lhs =
let f_vars_applied =
List.map (fun f_var -> T.app (T.var f_var) lhs_args) f_vars
in
T.app_builtin ~ty:(T.ty t) b (ty_params @ f_vars_applied)
in
let subst = US.FO.bind subst (v,sc) (lambda,sc) in
Iter.return ([ty_args@env, lhs, rhs],subst,offset,"imitate_b")
| T.Const _ ->
let t_mono = T.head_term_mono t in
let n, ty_args_t, _ = Type.open_poly_fun (T.ty t_mono) in
assert (n=0);
let offset, f_vars =
ty_args_t
|> List.map (Type.arrow all_ty_args)
|> mk_fresh_vars offset
in
let lambda =
let f_vars_applied =
List.map (fun f_var -> T.app (T.var f_var) all_vars) f_vars
in
T.app t_mono f_vars_applied
|> T.fun_l (List.map T.ty all_vars)
and lhs =
let f_vars_applied =
List.map (fun f_var -> T.app (T.var f_var) lhs_args) f_vars
in
T.app t_mono f_vars_applied
in
let subst = US.FO.bind subst (v,sc) (lambda,sc) in
Iter.return ([ty_args@env,lhs,rhs],subst,offset,"imitate")
| _ ->
Iter.empty
in
Iter.append imitate proj
let unif_loop (st:state): unit =
let sc = st.sc in
let add_sol subst penalty =
Util.debugf ~section 5 "(@[add_sol@ :subst %a@])" (fun k->k US.pp subst);
st.sols <- (subst, penalty) :: st.sols
in
while st.fuel > 0 && not (Queue.is_empty st.queue) do
let pb = Queue.pop st.queue in
let pb = normalize_pb sc pb in
begin match pb with
| None -> ()
| Some {pairs=[]; subst; penalty; _} ->
add_sol subst penalty;
| Some ({penalty; offset; subst; pairs=(env,t1,t2) :: pairs_tl} as pb) ->
Util.debugf ~section 5 "(@[ho_unif.try_pair %a@ :subst %a@])"
(fun k->k pp_pair (env,t1,t2) US.pp subst);
begin
try
let fail() = raise Unif.Fail in
let consume_fuel() = st.fuel <- st.fuel - 1 in
let push_new ~penalty ~subst ~offset rule pairs : unit =
let subst =
List.fold_left
(fun subst (_,t,u) ->
Unif.Ty.unify_full ~subst (T.ty t,sc) (T.ty u,sc))
subst pairs
in
let pb' =
mk_pb ~penalty ~subst ~offset (pairs @ pairs_tl)
|> normalize_pb sc
in
begin match pb' with
| None -> ()
| Some pb' ->
Util.debugf ~section 5 "(@[ho_unif.push@ :rule %s@ %a@])"
(fun k->k rule pp_pb pb');
Queue.push pb' st.queue
end
in
let subst = Unif.Ty.unify_full ~subst (T.ty t1,sc) (T.ty t2,sc) in
let t1 = whnf_deref subst (t1,sc) in
let t2 = whnf_deref subst (t2,sc) in
let hd1, l1 = T.as_app t1 in
let hd2, l2 = T.as_app t2 in
begin match T.view hd1, T.view hd2 with
| _ when T.equal t1 t2 ->
push_new "triv" ~penalty ~offset ~subst []
| T.Const id1, T.Const id2 ->
if ID.equal id1 id2 && List.length l1=List.length l2
then (
let new_pairs = mk_pairs env l1 l2 in
push_new "rigid" ~penalty ~offset ~subst new_pairs
) else fail()
| T.DB a, T.DB b ->
if a=b then push_new "db" ~penalty ~offset ~subst [] else fail()
| T.Var _, _ when l1=[] ->
let subst = Unif.FO.unify_full ~subst (t1,0) (t2,0) in
push_new "bind" ~penalty ~offset ~subst []
| _, T.Var _ when l2=[] ->
let subst = Unif.FO.unify_full ~subst (t1,0) (t2,0) in
push_new "bind" ~penalty ~offset ~subst []
| T.AppBuiltin (b1, l1'), T.AppBuiltin (b2,l2') ->
assert (l1=[]);
assert (l2=[]);
if Builtin.equal b1 b2 && List.length l1'=List.length l2'
then (
let new_pairs = mk_pairs env l1 l2 in
push_new "rigid" ~penalty ~offset ~subst new_pairs
) else fail()
| T.Var v, T.Fun _ ->
assert (l2=[]);
let pairs, offset = unif_lambda ~offset env v l1 hd2 in
push_new "bind" ~penalty ~subst ~offset pairs
| T.Fun _, T.Var v ->
assert (l1=[]);
let pairs, offset = unif_lambda ~offset env v l2 hd1 in
push_new "bind" ~penalty ~subst ~offset pairs
| T.Fun _, T.Fun _ ->
assert (l1 = []);
assert (l2 = []);
let n_params, ty_args, _ = Type.open_poly_fun (T.ty t1) in
assert (n_params=0);
let n = List.length ty_args in
let args = List.mapi (fun i ty -> T.bvar ~ty (n-i-1)) ty_args in
let pair = ty_args @ env, T.app hd1 args, T.app hd2 args in
push_new "eta" ~penalty ~subst ~offset [pair]
| T.Var v, (T.Const _ | T.AppBuiltin _ | T.DB _) ->
assert (l1<>[]);
consume_fuel();
unif_rigid ~sc ~subst ~offset env v l1 t2
(fun (pairs,subst,offset,rule) ->
push_new rule ~penalty ~subst ~offset pairs);
| (T.Const _ | T.AppBuiltin _ | T.DB _), T.Var v ->
assert (l2<>[]);
consume_fuel();
unif_rigid ~sc ~subst ~offset env v l2 t1
(fun (pairs,subst,offset,rule) ->
push_new rule ~penalty ~subst ~offset pairs);
| T.Var _, T.Var _ ->
Util.debugf ~section 5
"(@[ho_unif.all_flex_flex@ %a@])"
(fun k->k pp_pb pb);
assert (List.for_all is_flex_flex pairs_tl);
let subst =
List.fold_left
(fun subst p -> US.add_constr (delay_pair p sc) subst)
pb.subst pb.pairs
in
add_sol subst pb.penalty
| T.App _, _ | _, T.App _ -> assert false
| T.Const _, _
| T.Fun _, _
| T.AppBuiltin _, _
| T.DB _, _
-> fail()
end
with Unif.Fail ->
Util.debugf ~section 5 "(@[ho_unif.drop_pb@ %a@])"
(fun k->k pp_pb pb);
()
end
end
done
let norm_subst_ offset sc (us:US.t) () : US.t =
US.map_subst us
~f:(fun subst ->
Subst.normalize subst
|> Subst.FO.filter
(fun (v,sc_v) (t,sc_t) ->
let is_fvar (v,sc_v) =
sc_v = sc && HVar.id v >= offset &&
not (Type.is_tType (HVar.ty v))
in
not (is_fvar (v,sc_v)) ||
(T.Seq.vars t
|> Iter.exists (fun v' -> not (is_fvar (v',sc_t)))))
|> Subst.FO.map Lambda.snf
)
let norm_subst offset sc us =
if !enable_norm_subst
then Util.with_prof prof_norm_subst (norm_subst_ offset sc us) ()
else us
let apply_subst pairs us =
let renaming = Subst.Renaming.create() in
let subst = Unif_subst.subst us in
let pairs =
List.map
(fun (env,t,u) ->
let t = Subst.FO.apply renaming subst (T.fun_l env t,0) in
let u = Subst.FO.apply renaming subst (T.fun_l env u,0) in
[], t, u)
pairs
in
pairs, renaming
let get_solutions (st:state): (pair list * US.t * penalty * Subst.Renaming.t) list =
let sols1 =
st.sols
|> List.rev_map (fun (subst,p) -> [], norm_subst st.offset0 st.sc subst, p, Subst.Renaming.create())
and sols2 =
st.queue
|> Iter.of_queue
|> Iter.map
(fun pb ->
let pairs, renaming = apply_subst pb.pairs pb.subst in
pairs, norm_subst st.offset0 st.sc pb.subst, pb.penalty, renaming)
|> Iter.to_rev_list
in
List.rev_append sols1 sols2
end
let unif_pairs ?(fuel= !default_fuel) (pairs,sc) ~offset : _ list =
let st = U.empty sc fuel offset in
U.add st (U.mk_pb ~offset ~penalty:0 ~subst:US.empty pairs);
U.unif_loop st;
U.get_solutions st
let () =
Options.add_opts
[ "--ho-unif-fuel", Arg.Set_int default_fuel, " default amount of fuel for HO unification";
"--ho-unif-norm", Arg.Set enable_norm_subst, " normalize substitutions in HO unif";
"--no-ho-unif-norm", Arg.Clear enable_norm_subst, " do not normalize substitutions in HO unif";
]