(* Title: HOL/Tools/SMT/smt_translate.ML
Author: Sascha Boehme, TU Muenchen
Translate theorems into an SMT intermediate format and serialize them.
*)
signature SMT_TRANSLATE =
sig
(*intermediate term structure*)
datatype squant = SForall | SExists
datatype 'a spattern = SPat of 'a list | SNoPat of 'a list
datatype sterm =
SVar of int |
SApp of string * sterm list |
SLet of string * sterm * sterm |
SQua of squant * string list * sterm spattern list * sterm
(*translation configuration*)
type sign = {
logic: string,
sorts: string list,
dtyps: (string * (string * (string * string) list) list) list list,
funcs: (string * (string list * string)) list }
type config = {
logic: term list -> string,
has_datatypes: bool,
serialize: (string * string) list -> string list -> sign -> sterm list -> string }
type replay_data = {
context: Proof.context,
typs: typ Symtab.table,
terms: term Symtab.table,
ll_defs: term list,
rewrite_rules: thm list,
assms: (int * thm) list }
(*translation*)
val add_config: SMT_Util.class * (Proof.context -> config) -> Context.generic -> Context.generic
val translate: Proof.context -> (string * string) list -> string list -> (int * thm) list ->
string * replay_data
end;
structure SMT_Translate: SMT_TRANSLATE =
struct
(* intermediate term structure *)
datatype squant = SForall | SExists
datatype 'a spattern = SPat of 'a list | SNoPat of 'a list
datatype sterm =
SVar of int |
SApp of string * sterm list |
SLet of string * sterm * sterm |
SQua of squant * string list * sterm spattern list * sterm
(* translation configuration *)
type sign = {
logic: string,
sorts: string list,
dtyps: (string * (string * (string * string) list) list) list list,
funcs: (string * (string list * string)) list }
type config = {
logic: term list -> string,
has_datatypes: bool,
serialize: (string * string) list -> string list -> sign -> sterm list -> string }
type replay_data = {
context: Proof.context,
typs: typ Symtab.table,
terms: term Symtab.table,
ll_defs: term list,
rewrite_rules: thm list,
assms: (int * thm) list }
(* translation context *)
fun add_components_of_typ (Type (s, Ts)) =
cons (Long_Name.base_name s) #> fold_rev add_components_of_typ Ts
| add_components_of_typ (TFree (s, _)) = cons (perhaps (try (unprefix "'")) s)
| add_components_of_typ _ = I;
fun suggested_name_of_typ T = space_implode "_" (add_components_of_typ T []);
fun suggested_name_of_term (Const (s, _)) = Long_Name.base_name s
| suggested_name_of_term (Free (s, _)) = s
| suggested_name_of_term _ = Name.uu
val empty_tr_context = (Name.context, Typtab.empty, Termtab.empty)
val safe_suffix = "$"
fun add_typ T proper (cx as (names, typs, terms)) =
(case Typtab.lookup typs T of
SOME (name, _) => (name, cx)
| NONE =>
let
val sugg = Name.desymbolize (SOME true) (suggested_name_of_typ T) ^ safe_suffix
val (name, names') = Name.variant sugg names
val typs' = Typtab.update (T, (name, proper)) typs
in (name, (names', typs', terms)) end)
fun add_fun t sort (cx as (names, typs, terms)) =
(case Termtab.lookup terms t of
SOME (name, _) => (name, cx)
| NONE =>
let
val sugg = Name.desymbolize (SOME false) (suggested_name_of_term t) ^ safe_suffix
val (name, names') = Name.variant sugg names
val terms' = Termtab.update (t, (name, sort)) terms
in (name, (names', typs, terms')) end)
fun sign_of logic dtyps (_, typs, terms) = {
logic = logic,
sorts = Typtab.fold (fn (_, (n, true)) => cons n | _ => I) typs [],
dtyps = dtyps,
funcs = Termtab.fold (fn (_, (n, SOME ss)) => cons (n,ss) | _ => I) terms []}
fun replay_data_of ctxt ll_defs rules assms (_, typs, terms) =
let
fun add_typ (T, (n, _)) = Symtab.update (n, T)
val typs' = Typtab.fold add_typ typs Symtab.empty
fun add_fun (t, (n, _)) = Symtab.update (n, t)
val terms' = Termtab.fold add_fun terms Symtab.empty
in
{context = ctxt, typs = typs', terms = terms', ll_defs = ll_defs, rewrite_rules = rules,
assms = assms}
end
(* preprocessing *)
(** datatype declarations **)
fun collect_datatypes_and_records (tr_context, ctxt) ts =
let
val (declss, ctxt') = fold (Term.fold_types SMT_Datatypes.add_decls) ts ([], ctxt)
fun is_decl_typ T = exists (exists (equal T o fst)) declss
fun add_typ' T proper =
(case SMT_Builtin.dest_builtin_typ ctxt' T of
SOME n => pair n
| NONE => add_typ T proper)
fun tr_select sel =
let val T = Term.range_type (Term.fastype_of sel)
in add_fun sel NONE ##>> add_typ' T (not (is_decl_typ T)) end
fun tr_constr (constr, selects) =
add_fun constr NONE ##>> fold_map tr_select selects
fun tr_typ (T, cases) = add_typ' T false ##>> fold_map tr_constr cases
val (declss', tr_context') = fold_map (fold_map tr_typ) declss tr_context
fun add (constr, selects) =
Termtab.update (constr, length selects) #>
fold (Termtab.update o rpair 1) selects
val funcs = fold (fold (fold add o snd)) declss Termtab.empty
in ((funcs, declss', tr_context', ctxt'), ts) end
(* FIXME: also return necessary datatype and record theorems *)
(** eta-expand quantifiers, let expressions and built-ins *)
local
fun eta f T t = Abs (Name.uu, T, f (Term.incr_boundvars 1 t $ Bound 0))
fun exp f T = eta f (Term.domain_type (Term.domain_type T))
fun exp2 T q =
let val U = Term.domain_type T
in Abs (Name.uu, U, q $ eta I (Term.domain_type U) (Bound 0)) end
fun expf k i T t =
let val Ts = drop i (fst (SMT_Util.dest_funT k T))
in
Term.incr_boundvars (length Ts) t
|> fold_rev (fn i => fn u => u $ Bound i) (0 upto length Ts - 1)
|> fold_rev (fn T => fn u => Abs (Name.uu, T, u)) Ts
end
in
fun eta_expand ctxt funcs =
let
fun exp_func t T ts =
(case Termtab.lookup funcs t of
SOME k => Term.list_comb (t, ts) |> k <> length ts ? expf k (length ts) T
| NONE => Term.list_comb (t, ts))
fun expand ((q as Const (@{const_name All}, _)) $ Abs a) = q $ abs_expand a
| expand ((q as Const (@{const_name All}, T)) $ t) = q $ exp expand T t
| expand (q as Const (@{const_name All}, T)) = exp2 T q
| expand ((q as Const (@{const_name Ex}, _)) $ Abs a) = q $ abs_expand a
| expand ((q as Const (@{const_name Ex}, T)) $ t) = q $ exp expand T t
| expand (q as Const (@{const_name Ex}, T)) = exp2 T q
| expand (Const (@{const_name Let}, _) $ t $ u) = expand (Term.betapply (u, t))
| expand (Const (@{const_name Let}, T) $ t) =
let val U = Term.domain_type (Term.range_type T)
in Abs (Name.uu, U, Bound 0 $ Term.incr_boundvars 1 t) end
| expand (Const (@{const_name Let}, T)) =
let val U = Term.domain_type (Term.range_type T)
in Abs (Name.uu, Term.domain_type T, Abs (Name.uu, U, Bound 0 $ Bound 1)) end
| expand t =
(case Term.strip_comb t of
(u as Const (c as (_, T)), ts) =>
(case SMT_Builtin.dest_builtin ctxt c ts of
SOME (_, k, us, mk) =>
if k = length us then mk (map expand us)
else if k < length us then chop k (map expand us) |>> mk |> Term.list_comb
else expf k (length ts) T (mk (map expand us))
| NONE => exp_func u T (map expand ts))
| (u as Free (_, T), ts) => exp_func u T (map expand ts)
| (Abs a, ts) => Term.list_comb (abs_expand a, map expand ts)
| (u, ts) => Term.list_comb (u, map expand ts))
and abs_expand (n, T, t) = Abs (n, T, expand t)
in map expand end
end
(** introduce explicit applications **)
local
(*
Make application explicit for functions with varying number of arguments.
*)
fun add t i = apfst (Termtab.map_default (t, i) (Integer.min i))
fun add_type T = apsnd (Typtab.update (T, ()))
fun min_arities t =
(case Term.strip_comb t of
(u as Const _, ts) => add u (length ts) #> fold min_arities ts
| (u as Free _, ts) => add u (length ts) #> fold min_arities ts
| (Abs (_, T, u), ts) => (can dest_funT T ? add_type T) #> min_arities u #> fold min_arities ts
| (_, ts) => fold min_arities ts)
fun minimize types t i =
let
fun find_min j [] _ = j
| find_min j (U :: Us) T =
if Typtab.defined types T then j else find_min (j + 1) Us (U --> T)
val (Ts, T) = Term.strip_type (Term.type_of t)
in find_min 0 (take i (rev Ts)) T end
fun app u (t, T) = (Const (@{const_name fun_app}, T --> T) $ t $ u, Term.range_type T)
fun apply i t T ts =
let
val (ts1, ts2) = chop i ts
val (_, U) = SMT_Util.dest_funT i T
in fst (fold app ts2 (Term.list_comb (t, ts1), U)) end
in
fun intro_explicit_application ctxt funcs ts =
let
val (arities, types) = fold min_arities ts (Termtab.empty, Typtab.empty)
val arities' = Termtab.map (minimize types) arities (* FIXME: highly suspicious *)
fun app_func t T ts =
if is_some (Termtab.lookup funcs t) then Term.list_comb (t, ts)
else apply (the (Termtab.lookup arities' t)) t T ts
fun in_list T f t = SMT_Util.mk_symb_list T (map f (SMT_Util.dest_symb_list t))
fun traverse Ts t =
(case Term.strip_comb t of
(q as Const (@{const_name All}, _), [Abs (x, T, u)]) =>
q $ Abs (x, T, in_trigger (T :: Ts) u)
| (q as Const (@{const_name Ex}, _), [Abs (x, T, u)]) =>
q $ Abs (x, T, in_trigger (T :: Ts) u)
| (q as Const (@{const_name Let}, _), [u1, u2 as Abs _]) =>
q $ traverse Ts u1 $ traverse Ts u2
| (u as Const (c as (_, T)), ts) =>
(case SMT_Builtin.dest_builtin ctxt c ts of
SOME (_, k, us, mk) =>
let
val (ts1, ts2) = chop k (map (traverse Ts) us)
val U = Term.strip_type T |>> snd o chop k |> (op --->)
in apply 0 (mk ts1) U ts2 end
| NONE => app_func u T (map (traverse Ts) ts))
| (u as Free (_, T), ts) => app_func u T (map (traverse Ts) ts)
| (u as Bound i, ts) => apply 0 u (nth Ts i) (map (traverse Ts) ts)
| (Abs (n, T, u), ts) => traverses Ts (Abs (n, T, traverse (T::Ts) u)) ts
| (u, ts) => traverses Ts u ts)
and in_trigger Ts ((c as @{const trigger}) $ p $ t) = c $ in_pats Ts p $ traverse Ts t
| in_trigger Ts t = traverse Ts t
and in_pats Ts ps =
in_list @{typ "pattern symb_list"} (in_list @{typ pattern} (in_pat Ts)) ps
and in_pat Ts ((p as Const (@{const_name pat}, _)) $ t) = p $ traverse Ts t
| in_pat Ts ((p as Const (@{const_name nopat}, _)) $ t) = p $ traverse Ts t
| in_pat _ t = raise TERM ("bad pattern", [t])
and traverses Ts t ts = Term.list_comb (t, map (traverse Ts) ts)
in map (traverse []) ts end
val fun_app_eq = mk_meta_eq @{thm fun_app_def}
end
(** map HOL formulas to FOL formulas (i.e., separate formulas froms terms) **)
local
val is_quant = member (op =) [@{const_name All}, @{const_name Ex}]
val fol_rules = [
Let_def,
@{lemma "P = True == P" by (rule eq_reflection) simp},
@{lemma "if P then True else False == P" by (rule eq_reflection) simp}]
exception BAD_PATTERN of unit
fun wrap_in_if pat t =
if pat then raise BAD_PATTERN () else @{const If (bool)} $ t $ @{const True} $ @{const False}
fun is_builtin_conn_or_pred ctxt c ts =
is_some (SMT_Builtin.dest_builtin_conn ctxt c ts) orelse
is_some (SMT_Builtin.dest_builtin_pred ctxt c ts)
in
fun folify ctxt =
let
fun in_list T f t = SMT_Util.mk_symb_list T (map_filter f (SMT_Util.dest_symb_list t))
fun in_term pat t =
(case Term.strip_comb t of
(@{const True}, []) => t
| (@{const False}, []) => t
| (u as Const (@{const_name If}, _), [t1, t2, t3]) =>
if pat then raise BAD_PATTERN () else u $ in_form t1 $ in_term pat t2 $ in_term pat t3
| (Const (c as (n, _)), ts) =>
if is_builtin_conn_or_pred ctxt c ts then wrap_in_if pat (in_form t)
else if is_quant n then wrap_in_if pat (in_form t)
else Term.list_comb (Const c, map (in_term pat) ts)
| (Free c, ts) => Term.list_comb (Free c, map (in_term pat) ts)
| _ => t)
and in_pat ((p as Const (@{const_name pat}, _)) $ t) =
p $ in_term true t
| in_pat ((p as Const (@{const_name nopat}, _)) $ t) =
p $ in_term true t
| in_pat t = raise TERM ("bad pattern", [t])
and in_pats ps =
in_list @{typ "pattern symb_list"} (SOME o in_list @{typ pattern} (try in_pat)) ps
and in_trigger ((c as @{const trigger}) $ p $ t) = c $ in_pats p $ in_form t
| in_trigger t = in_form t
and in_form t =
(case Term.strip_comb t of
(q as Const (qn, _), [Abs (n, T, u)]) =>
if is_quant qn then q $ Abs (n, T, in_trigger u)
else in_term false t
| (Const c, ts) =>
(case SMT_Builtin.dest_builtin_conn ctxt c ts of
SOME (_, _, us, mk) => mk (map in_form us)
| NONE =>
(case SMT_Builtin.dest_builtin_pred ctxt c ts of
SOME (_, _, us, mk) => mk (map (in_term false) us)
| NONE => in_term false t))
| _ => in_term false t)
in
map in_form #>
pair (fol_rules, I)
end
end
(* translation into intermediate format *)
(** utility functions **)
val quantifier = (fn
@{const_name All} => SOME SForall
| @{const_name Ex} => SOME SExists
| _ => NONE)
fun group_quant qname Ts (t as Const (q, _) $ Abs (_, T, u)) =
if q = qname then group_quant qname (T :: Ts) u else (Ts, t)
| group_quant _ Ts t = (Ts, t)
fun dest_pat (Const (@{const_name pat}, _) $ t) = (t, true)
| dest_pat (Const (@{const_name nopat}, _) $ t) = (t, false)
| dest_pat t = raise TERM ("bad pattern", [t])
fun dest_pats [] = I
| dest_pats ts =
(case map dest_pat ts |> split_list ||> distinct (op =) of
(ps, [true]) => cons (SPat ps)
| (ps, [false]) => cons (SNoPat ps)
| _ => raise TERM ("bad multi-pattern", ts))
fun dest_trigger (@{const trigger} $ tl $ t) =
(rev (fold (dest_pats o SMT_Util.dest_symb_list) (SMT_Util.dest_symb_list tl) []), t)
| dest_trigger t = ([], t)
fun dest_quant qn T t = quantifier qn |> Option.map (fn q =>
let
val (Ts, u) = group_quant qn [T] t
val (ps, p) = dest_trigger u
in (q, rev Ts, ps, p) end)
fun fold_map_pat f (SPat ts) = fold_map f ts #>> SPat
| fold_map_pat f (SNoPat ts) = fold_map f ts #>> SNoPat
(** translation from Isabelle terms into SMT intermediate terms **)
fun intermediate logic dtyps builtin ctxt ts trx =
let
fun transT (T as TFree _) = add_typ T true
| transT (T as TVar _) = (fn _ => raise TYPE ("bad SMT type", [T], []))
| transT (T as Type _) =
(case SMT_Builtin.dest_builtin_typ ctxt T of
SOME n => pair n
| NONE => add_typ T true)
fun app n ts = SApp (n, ts)
fun trans t =
(case Term.strip_comb t of
(Const (qn, _), [Abs (_, T, t1)]) =>
(case dest_quant qn T t1 of
SOME (q, Ts, ps, b) =>
fold_map transT Ts ##>> fold_map (fold_map_pat trans) ps ##>>
trans b #>> (fn ((Ts', ps'), b') => SQua (q, Ts', ps', b'))
| NONE => raise TERM ("unsupported quantifier", [t]))
| (Const (@{const_name Let}, _), [t1, Abs (_, T, t2)]) =>
transT T ##>> trans t1 ##>> trans t2 #>> (fn ((U, u1), u2) => SLet (U, u1, u2))
| (u as Const (c as (_, T)), ts) =>
(case builtin ctxt c ts of
SOME (n, _, us, _) => fold_map trans us #>> app n
| NONE => transs u T ts)
| (u as Free (_, T), ts) => transs u T ts
| (Bound i, []) => pair (SVar i)
| _ => raise TERM ("bad SMT term", [t]))
and transs t T ts =
let val (Us, U) = SMT_Util.dest_funT (length ts) T
in
fold_map transT Us ##>> transT U #-> (fn Up =>
add_fun t (SOME Up) ##>> fold_map trans ts #>> SApp)
end
val (us, trx') = fold_map trans ts trx
in ((sign_of (logic ts) dtyps trx', us), trx') end
(* translation *)
structure Configs = Generic_Data
(
type T = (Proof.context -> config) SMT_Util.dict
val empty = []
val extend = I
fun merge data = SMT_Util.dict_merge fst data
)
fun add_config (cs, cfg) = Configs.map (SMT_Util.dict_update (cs, cfg))
fun get_config ctxt =
let val cs = SMT_Config.solver_class_of ctxt
in
(case SMT_Util.dict_get (Configs.get (Context.Proof ctxt)) cs of
SOME cfg => cfg ctxt
| NONE => error ("SMT: no translation configuration found " ^
"for solver class " ^ quote (SMT_Util.string_of_class cs)))
end
fun translate ctxt smt_options comments ithms =
let
val {logic, has_datatypes, serialize} = get_config ctxt
fun no_dtyps (tr_context, ctxt) ts =
((Termtab.empty, [], tr_context, ctxt), ts)
val ts1 = map (Envir.beta_eta_contract o SMT_Util.prop_of o snd) ithms
val ((funcs, dtyps, tr_context, ctxt1), ts2) =
((empty_tr_context, ctxt), ts1)
|-> (if has_datatypes then collect_datatypes_and_records else no_dtyps)
fun is_binder (Const (@{const_name Let}, _) $ _) = true
| is_binder t = Lambda_Lifting.is_quantifier t
fun mk_trigger ((q as Const (@{const_name All}, _)) $ Abs (n, T, t)) =
q $ Abs (n, T, mk_trigger t)
| mk_trigger (eq as (Const (@{const_name HOL.eq}, T) $ lhs $ _)) =
Term.domain_type T --> @{typ pattern}
|> (fn T => Const (@{const_name pat}, T) $ lhs)
|> SMT_Util.mk_symb_list @{typ pattern} o single
|> SMT_Util.mk_symb_list @{typ "pattern symb_list"} o single
|> (fn t => @{const trigger} $ t $ eq)
| mk_trigger t = t
val (ctxt2, (ts3, ll_defs)) =
ts2
|> eta_expand ctxt1 funcs
|> rpair ctxt1
|-> Lambda_Lifting.lift_lambdas NONE is_binder
|-> (fn (ts', ll_defs) => fn ctxt' =>
(ctxt', (intro_explicit_application ctxt' funcs (map mk_trigger ll_defs @ ts'), ll_defs)))
val ((rewrite_rules, builtin), ts4) = folify ctxt2 ts3
|>> apfst (cons fun_app_eq)
in
(ts4, tr_context)
|-> intermediate logic dtyps (builtin SMT_Builtin.dest_builtin) ctxt2
|>> uncurry (serialize smt_options comments)
||> replay_data_of ctxt2 ll_defs rewrite_rules ithms
end
end;