(* 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 * int option * sterm
(*translation configuration*)
type prefixes = {sort_prefix: string, func_prefix: string}
type sign = {
header: string list,
sorts: string list,
dtyps: (string * (string * (string * string) list) list) list list,
funcs: (string * (string list * string)) list }
type config = {
prefixes: prefixes,
header: term list -> string list,
is_fol: bool,
has_datatypes: bool,
serialize: string list -> sign -> sterm list -> string }
type recon = {
context: Proof.context,
typs: typ Symtab.table,
terms: term Symtab.table,
rewrite_rules: thm list,
assms: (int * thm) list }
(*translation*)
val add_config: SMT_Utils.class * (Proof.context -> config) ->
Context.generic -> Context.generic
val translate: Proof.context -> string list -> (int * thm) list ->
string * recon
end
structure SMT_Translate: SMT_TRANSLATE =
struct
structure U = SMT_Utils
structure B = SMT_Builtin
(* 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 * int option * sterm
(* translation configuration *)
type prefixes = {sort_prefix: string, func_prefix: string}
type sign = {
header: string list,
sorts: string list,
dtyps: (string * (string * (string * string) list) list) list list,
funcs: (string * (string list * string)) list }
type config = {
prefixes: prefixes,
header: term list -> string list,
is_fol: bool,
has_datatypes: bool,
serialize: string list -> sign -> sterm list -> string }
type recon = {
context: Proof.context,
typs: typ Symtab.table,
terms: term Symtab.table,
rewrite_rules: thm list,
assms: (int * thm) list }
(* translation context *)
fun make_tr_context {sort_prefix, func_prefix} =
(sort_prefix, 1, Typtab.empty, func_prefix, 1, Termtab.empty)
fun string_of_index pre i = pre ^ string_of_int i
fun add_typ T proper (cx as (sp, Tidx, typs, fp, idx, terms)) =
(case Typtab.lookup typs T of
SOME (n, _) => (n, cx)
| NONE =>
let
val n = string_of_index sp Tidx
val typs' = Typtab.update (T, (n, proper)) typs
in (n, (sp, Tidx+1, typs', fp, idx, terms)) end)
fun add_fun t sort (cx as (sp, Tidx, typs, fp, idx, terms)) =
(case Termtab.lookup terms t of
SOME (n, _) => (n, cx)
| NONE =>
let
val n = string_of_index fp idx
val terms' = Termtab.update (t, (n, sort)) terms
in (n, (sp, Tidx, typs, fp, idx+1, terms')) end)
fun sign_of header dtyps (_, _, typs, _, _, terms) = {
header = header,
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 recon_of ctxt rules thms ithms revertT revert (_, _, typs, _, _, terms) =
let
fun add_typ (T, (n, _)) = Symtab.update (n, revertT T)
val typs' = Typtab.fold add_typ typs Symtab.empty
fun add_fun (t, (n, _)) = Symtab.update (n, revert t)
val terms' = Termtab.fold add_fun terms Symtab.empty
val assms = map (pair ~1) thms @ ithms
in
{context=ctxt, typs=typs', terms=terms', rewrite_rules=rules, assms=assms}
end
(* preprocessing *)
(** mark built-in constants as Var **)
fun mark_builtins ctxt =
let
(*
Note: schematic terms cannot occur anymore in terms at this stage.
*)
fun mark t =
(case Term.strip_comb t of
(u as Const (@{const_name If}, _), ts) => marks u ts
| (u as Const c, ts) =>
(case B.builtin_num ctxt t of
SOME (n, T) =>
let val v = ((n, 0), T)
in Vartab.update v #> pair (Var v) end
| NONE =>
(case B.builtin_fun ctxt c ts of
SOME ((ni, T), us, U) =>
Vartab.update (ni, U) #> marks (Var (ni, T)) us
| NONE => marks u ts))
| (Abs (n, T, u), ts) => mark u #-> (fn u' => marks (Abs (n, T, u')) ts)
| (u, ts) => marks u ts)
and marks t ts = fold_map mark ts #>> Term.list_comb o pair t
in (fn ts => swap (fold_map mark ts Vartab.empty)) end
fun mark_builtins' ctxt t = hd (snd (mark_builtins ctxt [t]))
(** FIXME **)
local
(*
mark constructors and selectors as Vars (forcing eta-expansion),
add missing datatype selectors via hypothetical definitions,
also return necessary datatype and record theorems
*)
in
fun collect_datatypes_and_records (tr_context, ctxt) ts =
(([], tr_context, ctxt), ts)
end
(** eta-expand quantifiers, let expressions and built-ins *)
local
fun eta T t = Abs (Name.uu, T, Term.incr_boundvars 1 t $ Bound 0)
fun exp T = eta (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 (Term.domain_type U) (Bound 0)) end
fun exp2' T l =
let val (U1, U2) = Term.dest_funT T ||> Term.domain_type
in Abs (Name.uu, U1, eta U2 (l $ Bound 0)) end
fun expf t i T ts =
let val Ts = U.dest_funT i T |> fst |> drop (length ts)
in Term.list_comb (t, ts) |> fold_rev eta Ts end
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 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 T t
| expand (q as Const (@{const_name Ex}, T)) = exp2 T q
| expand ((l as Const (@{const_name Let}, _)) $ t $ Abs a) =
l $ expand t $ abs_expand a
| expand ((l as Const (@{const_name Let}, T)) $ t $ u) =
l $ expand t $ exp (Term.range_type T) u
| expand ((l as Const (@{const_name Let}, T)) $ t) = exp2 T (l $ expand t)
| expand (l as Const (@{const_name Let}, T)) = exp2' T l
| expand (Abs a) = abs_expand a
| expand t =
(case Term.strip_comb t of
(u as Const (@{const_name If}, T), ts) => expf u 3 T (map expand ts)
| (u as Var ((_, i), T), ts) => expf u i T (map expand ts)
| (u, ts) => Term.list_comb (u, map expand ts))
and abs_expand (n, T, t) = Abs (n, T, expand t)
in
val eta_expand = map expand
end
(** lambda-lifting **)
local
fun mk_def Ts T lhs rhs =
let
val eq = HOLogic.eq_const T $ lhs $ rhs
val trigger =
[[Const (@{const_name SMT.pat}, T --> @{typ SMT.pattern}) $ lhs]]
|> map (HOLogic.mk_list @{typ SMT.pattern})
|> HOLogic.mk_list @{typ "SMT.pattern list"}
fun mk_all T t = HOLogic.all_const T $ Abs (Name.uu, T, t)
in fold mk_all Ts (@{const SMT.trigger} $ trigger $ eq) end
fun replace_lambda Us Ts t (cx as (defs, ctxt)) =
let
val T = Term.fastype_of1 (Us @ Ts, t)
val lev = length Us
val bs = sort int_ord (Term.add_loose_bnos (t, lev, []))
val bss = map_index (fn (i, j) => (j, Integer.add lev i)) bs
val norm = perhaps (AList.lookup (op =) bss)
val t' = Term.map_aterms (fn Bound i => Bound (norm i) | t => t) t
val Ts' = map (nth Ts) bs
fun mk_abs U u = Abs (Name.uu, U, u)
val abs_rhs = fold mk_abs Ts' (fold mk_abs Us t')
in
(case Termtab.lookup defs abs_rhs of
SOME (f, _) => (Term.list_comb (f, map Bound bs), cx)
| NONE =>
let
val (n, ctxt') =
yield_singleton Variable.variant_fixes Name.uu ctxt
val f = Free (n, rev Ts' ---> (rev Us ---> T))
fun mk_bapp i t = t $ Bound i
val lhs =
f
|> fold_rev (mk_bapp o snd) bss
|> fold_rev mk_bapp (0 upto (length Us - 1))
val def = mk_def (Us @ Ts') T lhs t'
in (f, (Termtab.update (abs_rhs, (f, def)) defs, ctxt')) end)
end
fun dest_abs Ts (Abs (_, T, t)) = dest_abs (T :: Ts) t
| dest_abs Ts t = (Ts, t)
fun traverse Ts t =
(case t of
(q as Const (@{const_name All}, _)) $ Abs a =>
abs_traverse Ts a #>> (fn a' => q $ Abs a')
| (q as Const (@{const_name Ex}, _)) $ Abs a =>
abs_traverse Ts a #>> (fn a' => q $ Abs a')
| (l as Const (@{const_name Let}, _)) $ u $ Abs a =>
traverse Ts u ##>> abs_traverse Ts a #>>
(fn (u', a') => l $ u' $ Abs a')
| Abs _ =>
let val (Us, u) = dest_abs [] t
in traverse (Us @ Ts) u #-> replace_lambda Us Ts end
| u1 $ u2 => traverse Ts u1 ##>> traverse Ts u2 #>> (op $)
| _ => pair t)
and abs_traverse Ts (n, T, t) = traverse (T::Ts) t #>> (fn t' => (n, T, t'))
in
fun lift_lambdas ctxt ts =
(Termtab.empty, ctxt)
|> fold_map (traverse []) ts
|> (fn (us, (defs, ctxt')) =>
(ctxt', Termtab.fold (cons o snd o snd) defs us))
end
(** introduce explicit applications **)
local
(*
Make application explicit for functions with varying number of arguments.
*)
fun add t ts =
Termtab.map_default (t, []) (Ord_List.insert int_ord (length ts))
fun collect t =
(case Term.strip_comb t of
(u as Const _, ts) => add u ts #> fold collect ts
| (u as Free _, ts) => add u ts #> fold collect ts
| (Abs (_, _, u), ts) => collect u #> fold collect ts
| (_, ts) => fold collect ts)
fun app ts (t, T) =
let val f = Const (@{const_name SMT.fun_app}, T --> T)
in (Term.list_comb (f $ t, ts), snd (U.dest_funT (length ts) T)) end
fun appl _ _ [] = fst
| appl _ [] ts = fst o app ts
| appl i (k :: ks) ts =
let val (ts1, ts2) = chop (k - i) ts
in appl k ks ts2 o app ts1 end
fun appl0 [_] ts (t, _) = Term.list_comb (t, ts)
| appl0 (0 :: ks) ts tT = appl 0 ks ts tT
| appl0 ks ts tT = appl 0 ks ts tT
fun apply terms T t ts = appl0 (Termtab.lookup_list terms t) ts (t, T)
fun get_arities i t =
(case Term.strip_comb t of
(Bound j, ts) =>
(if i = j then Ord_List.insert int_ord (length ts) else I) #>
fold (get_arities i) ts
| (Abs (_, _, u), ts) => get_arities (i+1) u #> fold (get_arities i) ts
| (_, ts) => fold (get_arities i) ts)
in
fun intro_explicit_application ts =
let
val terms = fold collect ts Termtab.empty
fun traverse (env as (arities, Ts)) t =
(case Term.strip_comb t of
(u as Const (_, T), ts) => apply terms T u (map (traverse env) ts)
| (u as Free (_, T), ts) => apply terms T u (map (traverse env) ts)
| (u as Bound i, ts) =>
appl0 (nth arities i) (map (traverse env) ts) (u, nth Ts i)
| (Abs (n, T, u), ts) =>
let val env' = (get_arities 0 u [] :: arities, T :: Ts)
in traverses env (Abs (n, T, traverse env' u)) ts end
| (u, ts) => traverses env u ts)
and traverses env t ts = Term.list_comb (t, map (traverse env) ts)
in map (traverse ([], [])) ts end
val fun_app_eq = mk_meta_eq @{thm SMT.fun_app_def}
end
(** map HOL formulas to FOL formulas (i.e., separate formulas froms terms) **)
val tboolT = @{typ SMT.term_bool}
val term_true = Const (@{const_name True}, tboolT)
val term_false = Const (@{const_name False}, tboolT)
val term_bool = @{lemma "True ~= False" by simp}
val term_bool_prop =
let
fun replace @{const HOL.eq (bool)} = @{const HOL.eq (SMT.term_bool)}
| replace @{const True} = term_true
| replace @{const False} = term_false
| replace t = t
in
Term.map_aterms replace (HOLogic.dest_Trueprop (Thm.prop_of term_bool))
end
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}]
fun reduce_let (Const (@{const_name Let}, _) $ t $ u) =
reduce_let (Term.betapply (u, t))
| reduce_let (t $ u) = reduce_let t $ reduce_let u
| reduce_let (Abs (n, T, t)) = Abs (n, T, reduce_let t)
| reduce_let t = t
fun is_pred_type NONE = false
| is_pred_type (SOME T) = (Term.body_type T = @{typ bool})
fun is_conn_type NONE = false
| is_conn_type (SOME T) =
forall (equal @{typ bool}) (Term.body_type T :: Term.binder_types T)
fun revert_typ @{typ SMT.term_bool} = @{typ bool}
| revert_typ (Type (n, Ts)) = Type (n, map revert_typ Ts)
| revert_typ T = T
val revert_types = Term.map_types revert_typ
fun folify ctxt builtins =
let
fun as_term t = @{const HOL.eq (SMT.term_bool)} $ t $ term_true
fun as_tbool @{typ bool} = tboolT
| as_tbool (Type (n, Ts)) = Type (n, map as_tbool Ts)
| as_tbool T = T
fun mapTs f g i = U.dest_funT i #> (fn (Ts, T) => map f Ts ---> g T)
fun predT i = mapTs as_tbool I i
fun funcT i = mapTs as_tbool as_tbool i
fun func i (n, T) = (n, funcT i T)
fun map_ifT T = T |> Term.dest_funT ||> funcT 2 |> (op -->)
val if_term = @{const If (bool)} |> Term.dest_Const ||> map_ifT |> Const
fun wrap_in_if t = if_term $ t $ term_true $ term_false
fun in_list T f t = HOLogic.mk_list T (map f (HOLogic.dest_list t))
fun in_term t =
(case Term.strip_comb t of
(Const (n as @{const_name If}, T), [t1, t2, t3]) =>
Const (n, map_ifT T) $ in_form t1 $ in_term t2 $ in_term t3
| (Const (@{const_name HOL.eq}, _), _) => wrap_in_if (in_form t)
| (Var (ni as (_, i), T), ts) =>
let val U = Vartab.lookup builtins ni
in
if is_conn_type U orelse is_pred_type U then wrap_in_if (in_form t)
else Term.list_comb (Var (ni, funcT i T), map in_term ts)
end
| (Const c, ts) =>
Term.list_comb (Const (func (length ts) c), map in_term ts)
| (Free c, ts) =>
Term.list_comb (Free (func (length ts) c), map in_term ts)
| _ => t)
and in_weight ((c as @{const SMT.weight}) $ w $ t) = c $ w $ in_form t
| in_weight t = in_form t
and in_pat (Const (c as (@{const_name pat}, _)) $ t) =
Const (func 1 c) $ in_term t
| in_pat (Const (c as (@{const_name nopat}, _)) $ t) =
Const (func 1 c) $ in_term t
| in_pat t = raise TERM ("bad pattern", [t])
and in_pats ps =
in_list @{typ "pattern list"} (in_list @{typ pattern} in_pat) ps
and in_trig ((c as @{const trigger}) $ p $ t) = c $ in_pats p $ in_weight t
| in_trig t = in_weight t
and in_form t =
(case Term.strip_comb t of
(q as Const (qn, _), [Abs (n, T, u)]) =>
if member (op =) [@{const_name All}, @{const_name Ex}] qn then
q $ Abs (n, as_tbool T, in_trig u)
else as_term (in_term t)
| (u as Const (@{const_name If}, _), ts) =>
Term.list_comb (u, map in_form ts)
| (b as @{const HOL.eq (bool)}, ts) => Term.list_comb (b, map in_form ts)
| (Const (n as @{const_name HOL.eq}, T), ts) =>
Term.list_comb (Const (n, predT 2 T), map in_term ts)
| (b as Var (ni as (_, i), T), ts) =>
if is_conn_type (Vartab.lookup builtins ni) then
Term.list_comb (b, map in_form ts)
else if is_pred_type (Vartab.lookup builtins ni) then
Term.list_comb (Var (ni, predT i T), map in_term ts)
else as_term (in_term t)
| _ => as_term (in_term t))
in
map (reduce_let #> in_form) #>
cons (mark_builtins' ctxt term_bool_prop) #>
pair (fol_rules, [term_bool])
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_weight (@{const SMT.weight} $ w $ t) =
((SOME (snd (HOLogic.dest_number w)), t)
handle TERM _ =>
(case w of
Var ((s, _), _) => (* FIXME: temporary workaround *)
(case Int.fromString s of
SOME n => (SOME n, t)
| NONE => raise TERM ("bad weight", [w]))
| _ => raise TERM ("bad weight", [w])))
| dest_weight t = (NONE, 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 HOLogic.dest_list) (HOLogic.dest_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
val (w, b) = dest_weight p
in (q, rev Ts, ps, w, b) 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 header dtyps 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 B.builtin_typ ctxt T of
SOME n => pair n
| NONE => add_typ T true)
val unmarked_builtins = [@{const_name If}, @{const_name HOL.eq}]
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, w, b) =>
fold_map transT Ts ##>> fold_map (fold_map_pat trans) ps ##>>
trans b #>> (fn ((Ts', ps'), b') => SQua (q, Ts', ps', w, 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))
| (Var ((n, _), _), ts) => fold_map trans ts #>> app n
| (u as Const (c as (n, T)), ts) =>
if member (op =) unmarked_builtins n then
(case B.builtin_fun ctxt c ts of
SOME (((m, _), _), us, _) => fold_map trans us #>> app m
| NONE => raise TERM ("not a built-in symbol", [t]))
else 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) = U.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 (header ts) dtyps trx', us), trx') end
(* translation *)
structure Configs = Generic_Data
(
type T = (Proof.context -> config) U.dict
val empty = []
val extend = I
fun merge xx = U.dict_merge fst xx
)
fun add_config (cs, cfg) = Configs.map (U.dict_update (cs, cfg))
fun translate ctxt comments ithms =
let
val cs = SMT_Config.solver_class_of ctxt
val {prefixes, is_fol, header, has_datatypes, serialize} =
(case U.dict_get (Configs.get (Context.Proof ctxt)) cs of
SOME cfg => cfg ctxt
| NONE => error ("SMT: no translation configuration found " ^
"for solver class " ^ quote (U.string_of_class cs)))
val with_datatypes =
has_datatypes andalso Config.get ctxt SMT_Config.datatypes
fun no_dtyps (tr_context, ctxt) ts = (([], tr_context, ctxt), ts)
val (builtins, ts1) =
ithms
|> map (HOLogic.dest_Trueprop o Thm.prop_of o snd)
|> map (Envir.eta_contract o Envir.beta_norm)
|> mark_builtins ctxt
val ((dtyps, tr_context, ctxt1), ts2) =
((make_tr_context prefixes, ctxt), ts1)
|-> (if with_datatypes then collect_datatypes_and_records else no_dtyps)
val (ctxt2, ts3) =
ts2
|> eta_expand
|> lift_lambdas ctxt1
||> intro_explicit_application
val ((rewrite_rules, extra_thms), ts4) =
(if is_fol then folify ctxt2 builtins else pair ([], [])) ts3
val rewrite_rules' = fun_app_eq :: rewrite_rules
in
(ts4, tr_context)
|-> intermediate header dtyps ctxt2
|>> uncurry (serialize comments)
||> recon_of ctxt2 rewrite_rules' extra_thms ithms revert_typ revert_types
end
end