keep track of theorems initially given to SMT (even if they are rewritten); provide interface to filter theorems necessary for SMT proofs
(* 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
(* configuration options *)
type prefixes = {sort_prefix: string, func_prefix: string}
type header = Proof.context -> term list -> string list
type strict = {
is_builtin_conn: string * typ -> bool,
is_builtin_pred: Proof.context -> string * typ -> bool,
is_builtin_distinct: bool}
type builtins = {
builtin_typ: Proof.context -> typ -> string option,
builtin_num: Proof.context -> typ -> int -> string option,
builtin_fun: Proof.context -> string * typ -> term list ->
(string * term list) option,
has_datatypes: bool }
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: header,
strict: strict option,
builtins: builtins,
serialize: string list -> sign -> sterm list -> string }
type recon = {
typs: typ Symtab.table,
terms: term Symtab.table,
unfolds: thm list,
assms: (int * thm) list }
val translate: config -> Proof.context -> string list -> (int * thm) list ->
string * recon
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
(* configuration options *)
type prefixes = {sort_prefix: string, func_prefix: string}
type header = Proof.context -> term list -> string list
type strict = {
is_builtin_conn: string * typ -> bool,
is_builtin_pred: Proof.context -> string * typ -> bool,
is_builtin_distinct: bool}
type builtins = {
builtin_typ: Proof.context -> typ -> string option,
builtin_num: Proof.context -> typ -> int -> string option,
builtin_fun: Proof.context -> string * typ -> term list ->
(string * term list) option,
has_datatypes: bool }
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: header,
strict: strict option,
builtins: builtins,
serialize: string list -> sign -> sterm list -> string }
type recon = {
typs: typ Symtab.table,
terms: term Symtab.table,
unfolds: thm list,
assms: (int * thm) list }
(* utility functions *)
val dest_funT =
let
fun dest Ts 0 T = (rev Ts, T)
| dest Ts i (Type ("fun", [T, U])) = dest (T::Ts) (i-1) U
| dest _ _ T = raise TYPE ("dest_funT", [T], [])
in dest [] end
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 ("dest_pat", [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 ("dest_pats", ts))
fun dest_trigger (@{term 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, b) = dest_trigger u
in (q, rev Ts, ps, 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
fun prop_of thm = HOLogic.dest_Trueprop (Thm.prop_of thm)
(* enforce a strict separation between formulas and terms *)
val term_eq_rewr = @{lemma "term_eq x y == x = y" by (simp add: term_eq_def)}
val term_bool = @{lemma "~(term_eq True False)" by (simp add: term_eq_def)}
val term_bool' = Simplifier.rewrite_rule [term_eq_rewr] term_bool
val needs_rewrite = Thm.prop_of #> Term.exists_subterm (fn
Const (@{const_name Let}, _) => true
| @{term "op = :: bool => _"} $ _ $ @{term True} => true
| Const (@{const_name If}, _) $ _ $ @{term True} $ @{term False} => true
| _ => false)
val rewrite_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 rewrite ctxt = Simplifier.full_rewrite
(Simplifier.context ctxt empty_ss addsimps rewrite_rules)
fun normalize ctxt thm =
if needs_rewrite thm then Conv.fconv_rule (rewrite ctxt) thm else thm
val unfold_rules = term_eq_rewr :: rewrite_rules
val revert_types =
let
fun revert @{typ prop} = @{typ bool}
| revert (Type (n, Ts)) = Type (n, map revert Ts)
| revert T = T
in Term.map_types revert end
fun strictify {is_builtin_conn, is_builtin_pred, is_builtin_distinct} ctxt =
let
fun is_builtin_conn' (@{const_name True}, _) = false
| is_builtin_conn' (@{const_name False}, _) = false
| is_builtin_conn' c = is_builtin_conn c
val propT = @{typ prop} and boolT = @{typ bool}
val as_propT = (fn @{typ bool} => propT | T => T)
fun mapTs f g = Term.strip_type #> (fn (Ts, T) => map f Ts ---> g T)
fun conn (n, T) = (n, mapTs as_propT as_propT T)
fun pred (n, T) = (n, mapTs I as_propT T)
val term_eq = @{term "op = :: bool => _"} |> Term.dest_Const |> pred
fun as_term t = Const term_eq $ t $ @{term True}
val if_term = Const (@{const_name If}, [propT, boolT, boolT] ---> boolT)
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
(c as Const (@{const_name If}, _), [t1, t2, t3]) =>
c $ in_form t1 $ in_term t2 $ in_term t3
| (h as Const c, ts) =>
if is_builtin_conn' (conn c) orelse is_builtin_pred ctxt (pred c)
then wrap_in_if (in_form t)
else Term.list_comb (h, map in_term ts)
| (h as Free _, ts) => Term.list_comb (h, map in_term ts)
| _ => t)
and in_pat ((c as Const (@{const_name pat}, _)) $ t) = c $ in_term t
| in_pat ((c as Const (@{const_name nopat}, _)) $ t) = c $ in_term t
| in_pat t = raise TERM ("in_pat", [t])
and in_pats ps =
in_list @{typ "pattern list"} (in_list @{typ pattern} in_pat) ps
and in_trig ((c as @{term trigger}) $ p $ t) = c $ in_pats p $ in_form t
| in_trig t = in_form t
and in_form t =
(case Term.strip_comb t of
(q as Const (qn, _), [Abs (n, T, t')]) =>
if is_some (quantifier qn) then q $ Abs (n, T, in_trig t')
else as_term (in_term t)
| (Const (c as (@{const_name distinct}, T)), [t']) =>
if is_builtin_distinct then Const (pred c) $ in_list T in_term t'
else as_term (in_term t)
| (Const c, ts) =>
if is_builtin_conn (conn c)
then Term.list_comb (Const (conn c), map in_form ts)
else if is_builtin_pred ctxt (pred c)
then Term.list_comb (Const (pred c), map in_term ts)
else as_term (in_term t)
| _ => as_term (in_term t))
in
map (apsnd (normalize ctxt)) #> (fn irules =>
((unfold_rules, (~1, term_bool') :: irules),
map (in_form o prop_of o snd) ((~1, term_bool) :: irules)))
end
(* translation from Isabelle terms into SMT intermediate terms *)
val empty_context = (1, Typtab.empty, [], 1, Termtab.empty)
fun make_sign header (_, typs, dtyps, _, terms) = {
header = header,
sorts = Typtab.fold (fn (_, (n, true)) => cons n | _ => I) typs [],
funcs = Termtab.fold (fn (_, (n, SOME ss)) => cons (n,ss) | _ => I) terms [],
dtyps = rev dtyps }
fun make_recon (unfolds, assms) (_, typs, _, _, terms) = {
typs = Symtab.make (map (apfst fst o swap) (Typtab.dest typs)),
(*FIXME: don't drop the datatype information! *)
terms = Symtab.make (map (fn (t, (n, _)) => (n, t)) (Termtab.dest terms)),
unfolds = unfolds,
assms = assms }
fun string_of_index pre i = pre ^ string_of_int i
fun new_typ sort_prefix proper T (Tidx, typs, dtyps, idx, terms) =
let val s = string_of_index sort_prefix Tidx
in (s, (Tidx+1, Typtab.update (T, (s, proper)) typs, dtyps, idx, terms)) end
fun lookup_typ (_, typs, _, _, _) = Typtab.lookup typs
fun fresh_typ T f cx =
(case lookup_typ cx T of
SOME (s, _) => (s, cx)
| NONE => f T cx)
fun new_fun func_prefix t ss (Tidx, typs, dtyps, idx, terms) =
let
val f = string_of_index func_prefix idx
val terms' = Termtab.update (revert_types t, (f, ss)) terms
in (f, (Tidx, typs, dtyps, idx+1, terms')) end
fun fresh_fun func_prefix t ss (cx as (_, _, _, _, terms)) =
(case Termtab.lookup terms t of
SOME (f, _) => (f, cx)
| NONE => new_fun func_prefix t ss cx)
fun mk_type (_, Tfs) (d as Datatype.DtTFree _) = the (AList.lookup (op =) Tfs d)
| mk_type Ts (Datatype.DtType (n, ds)) = Type (n, map (mk_type Ts) ds)
| mk_type (Tds, _) (Datatype.DtRec i) = nth Tds i
fun mk_selector ctxt Ts T n (i, d) =
(case Datatype_Selectors.lookup_selector ctxt (n, i+1) of
NONE => raise Fail ("missing selector for datatype constructor " ^ quote n)
| SOME m => mk_type Ts d |> (fn U => (Const (m, T --> U), U)))
fun mk_constructor ctxt Ts T (n, args) =
let val (sels, Us) = split_list (map_index (mk_selector ctxt Ts T n) args)
in (Const (n, Us ---> T), sels) end
fun lookup_datatype ctxt n Ts =
if member (op =) [@{type_name bool}, @{type_name nat}] n then NONE
else
Datatype.get_info (ProofContext.theory_of ctxt) n
|> Option.map (fn {descr, ...} =>
let
val Tds = map (fn (_, (tn, _, _)) => Type (tn, Ts))
(sort (int_ord o pairself fst) descr)
val Tfs = (case hd descr of (_, (_, tfs, _)) => tfs ~~ Ts)
in
descr |> map (fn (i, (_, _, cs)) =>
(nth Tds i, map (mk_constructor ctxt (Tds, Tfs) (nth Tds i)) cs))
end)
fun relaxed irules = (([], irules), map (prop_of o snd) irules)
fun with_context header f (ths, ts) =
let val (us, context) = fold_map f ts empty_context
in ((make_sign (header ts) context, us), make_recon ths context) end
fun translate {prefixes, strict, header, builtins, serialize} ctxt comments =
let
val {sort_prefix, func_prefix} = prefixes
val {builtin_typ, builtin_num, builtin_fun, has_datatypes} = builtins
fun transT (T as TFree _) = fresh_typ T (new_typ sort_prefix true)
| transT (T as TVar _) = (fn _ => raise TYPE ("smt_translate", [T], []))
| transT (T as Type (n, Ts)) =
(case builtin_typ ctxt T of
SOME n => pair n
| NONE => fresh_typ T (fn _ => fn cx =>
if not has_datatypes then new_typ sort_prefix true T cx
else
(case lookup_datatype ctxt n Ts of
NONE => new_typ sort_prefix true T cx
| SOME dts =>
let val cx' = new_dtyps dts cx
in (fst (the (lookup_typ cx' T)), cx') end)))
and new_dtyps dts cx =
let
fun new_decl i t =
let val (Ts, T) = dest_funT i (Term.fastype_of t)
in
fold_map transT Ts ##>> transT T ##>>
new_fun func_prefix t NONE #>> swap
end
fun new_dtyp_decl (con, sels) =
new_decl (length sels) con ##>> fold_map (new_decl 1) sels #>>
(fn ((con', _), sels') => (con', map (apsnd snd) sels'))
in
cx
|> fold_map (new_typ sort_prefix false o fst) dts
||>> fold_map (fold_map new_dtyp_decl o snd) dts
|-> (fn (ss, decls) => fn (Tidx, typs, dtyps, idx, terms) =>
(Tidx, typs, (ss ~~ decls) :: dtyps, idx, terms))
end
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 ("intermediate", [t]))
| (Const (@{const_name Let}, _), [t1, Abs (_, T, t2)]) =>
transT T ##>> trans t1 ##>> trans t2 #>>
(fn ((U, u1), u2) => SLet (U, u1, u2))
| (h as Const (c as (@{const_name distinct}, T)), [t1]) =>
(case builtin_fun ctxt c (HOLogic.dest_list t1) of
SOME (n, ts) => fold_map trans ts #>> app n
| NONE => transs h T [t1])
| (h as Const (c as (_, T)), ts) =>
(case try HOLogic.dest_number t of
SOME (T, i) =>
(case builtin_num ctxt T i of
SOME n => pair (SApp (n, []))
| NONE => transs t T [])
| NONE =>
(case builtin_fun ctxt c ts of
SOME (n, ts') => fold_map trans ts' #>> app n
| NONE => transs h T ts))
| (h as Free (_, T), ts) => transs h T ts
| (Bound i, []) => pair (SVar i)
| _ => raise TERM ("smt_translate", [t]))
and transs t T ts =
let val (Us, U) = dest_funT (length ts) T
in
fold_map transT Us ##>> transT U #-> (fn Up =>
fresh_fun func_prefix t (SOME Up) ##>> fold_map trans ts #>> SApp)
end
in
(case strict of SOME strct => strictify strct ctxt | NONE => relaxed) #>
with_context (header ctxt) trans #>> uncurry (serialize comments)
end
end