(* Title: HOL/Tools/SMT2/smtlib2_proof.ML
Author: Sascha Boehme, TU Muenchen
Author: Mathias Fleury, ENS Rennes
Author: Jasmin Blanchette, TU Muenchen
SMT-LIB-2-style proofs: parsing and abstract syntax tree.
*)
signature SMTLIB2_PROOF =
sig
datatype 'b shared = Tree of SMTLIB2.tree | Term of term | Proof of 'b | None
type ('a, 'b) context
val mk_context: Proof.context -> int -> 'b shared Symtab.table -> typ Symtab.table ->
term Symtab.table -> 'a -> ('a, 'b) context
val empty_context: Proof.context -> typ Symtab.table -> term Symtab.table -> ('a list, 'b) context
val ctxt_of: ('a, 'b) context -> Proof.context
val lookup_binding: ('a, 'b) context -> string -> 'b shared
val update_binding: string * 'b shared -> ('a, 'b) context -> ('a, 'b) context
val with_bindings: (string * 'b shared) list -> (('a, 'b) context -> 'c * ('d, 'b) context) ->
('a, 'b) context -> 'c * ('d, 'b) context
(*type and term parsers*)
type type_parser = SMTLIB2.tree * typ list -> typ option
type term_parser = SMTLIB2.tree * term list -> term option
val add_type_parser: type_parser -> Context.generic -> Context.generic
val add_term_parser: term_parser -> Context.generic -> Context.generic
exception SMTLIB2_PARSE of string * SMTLIB2.tree
val declare_fun: string -> typ -> ((string * typ) list, 'a) context ->
((string * typ) list, 'a) context
val dest_binding: SMTLIB2.tree -> string * 'a shared
val type_of: ('a, 'b) context -> SMTLIB2.tree -> typ
val term_of: SMTLIB2.tree -> ((string * (string * typ)) list, 'a) context ->
term * ((string * (string * typ)) list, 'a) context
end;
structure SMTLIB2_Proof: SMTLIB2_PROOF =
struct
(* proof parser context *)
datatype 'b shared = Tree of SMTLIB2.tree | Term of term | Proof of 'b | None
type ('a, 'b) context = {
ctxt: Proof.context,
id: int,
syms: 'b shared Symtab.table,
typs: typ Symtab.table,
funs: term Symtab.table,
extra: 'a}
fun mk_context ctxt id syms typs funs extra: ('a, 'b) context =
{ctxt=ctxt, id=id, syms=syms, typs=typs, funs=funs, extra=extra}
fun empty_context ctxt typs funs = mk_context ctxt 1 Symtab.empty typs funs []
fun ctxt_of ({ctxt, ...}: ('a, 'b) context) = ctxt
fun lookup_binding ({syms, ...}: ('a, 'b) context) =
the_default None o Symtab.lookup syms
fun map_syms f ({ctxt, id, syms, typs, funs, extra}: ('a, 'b) context) =
mk_context ctxt id (f syms) typs funs extra
fun update_binding b = map_syms (Symtab.update b)
fun with_bindings bs f cx =
let val bs' = map (lookup_binding cx o fst) bs
in
cx
|> fold update_binding bs
|> f
||> fold2 (fn (name, _) => update_binding o pair name) bs bs'
end
fun next_id ({ctxt, id, syms, typs, funs, extra}: ('a, 'b) context) =
(id, mk_context ctxt (id + 1) syms typs funs extra)
fun with_fresh_names f ({ctxt, id, syms, typs, funs, extra}: ('a, 'b) context) =
let
fun bind (_, v as (_, T)) t = Logic.all_const T $ Term.absfree v t
val needs_inferT = equal Term.dummyT orf Term.is_TVar
val needs_infer = Term.exists_type (Term.exists_subtype needs_inferT)
fun infer_types ctxt =
singleton (Type_Infer_Context.infer_types ctxt) #>
singleton (Proof_Context.standard_term_check_finish ctxt)
fun infer ctxt t = if needs_infer t then infer_types ctxt t else t
type bindings = (string * (string * typ)) list
val (t, {ctxt=ctxt', extra=names, ...}: (bindings, 'b) context) =
f (mk_context ctxt id syms typs funs [])
val t' = infer ctxt' (fold_rev bind names (HOLogic.mk_Trueprop t))
in
((t', map fst names), mk_context ctxt id syms typs funs extra)
end
fun lookup_typ ({typs, ...}: ('a, 'b) context) = Symtab.lookup typs
fun lookup_fun ({funs, ...}: ('a, 'b) context) = Symtab.lookup funs
(* core type and term parser *)
fun core_type_parser (SMTLIB2.Sym "Bool", []) = SOME @{typ HOL.bool}
| core_type_parser (SMTLIB2.Sym "Int", []) = SOME @{typ Int.int}
| core_type_parser _ = NONE
fun mk_unary n t =
let val T = fastype_of t
in Const (n, T --> T) $ t end
fun mk_binary' n T U t1 t2 = Const (n, [T, T] ---> U) $ t1 $ t2
fun mk_binary n t1 t2 =
let val T = fastype_of t1
in mk_binary' n T T t1 t2 end
fun mk_rassoc f t ts =
let val us = rev (t :: ts)
in fold f (tl us) (hd us) end
fun mk_lassoc f t ts = fold (fn u1 => fn u2 => f u2 u1) ts t
fun mk_lassoc' n = mk_lassoc (mk_binary n)
fun mk_binary_pred n S t1 t2 =
let
val T1 = fastype_of t1
val T2 = fastype_of t2
val T =
if T1 <> Term.dummyT then T1
else if T2 <> Term.dummyT then T2
else TVar (("?a", serial ()), S)
in mk_binary' n T @{typ HOL.bool} t1 t2 end
fun mk_less t1 t2 = mk_binary_pred @{const_name ord_class.less} @{sort linorder} t1 t2
fun mk_less_eq t1 t2 = mk_binary_pred @{const_name ord_class.less_eq} @{sort linorder} t1 t2
fun core_term_parser (SMTLIB2.Sym "true", _) = SOME @{const HOL.True}
| core_term_parser (SMTLIB2.Sym "false", _) = SOME @{const HOL.False}
| core_term_parser (SMTLIB2.Sym "not", [t]) = SOME (HOLogic.mk_not t)
| core_term_parser (SMTLIB2.Sym "and", t :: ts) = SOME (mk_rassoc (curry HOLogic.mk_conj) t ts)
| core_term_parser (SMTLIB2.Sym "or", t :: ts) = SOME (mk_rassoc (curry HOLogic.mk_disj) t ts)
| core_term_parser (SMTLIB2.Sym "=>", [t1, t2]) = SOME (HOLogic.mk_imp (t1, t2))
| core_term_parser (SMTLIB2.Sym "implies", [t1, t2]) = SOME (HOLogic.mk_imp (t1, t2))
| core_term_parser (SMTLIB2.Sym "=", [t1, t2]) = SOME (HOLogic.mk_eq (t1, t2))
| core_term_parser (SMTLIB2.Sym "~", [t1, t2]) = SOME (HOLogic.mk_eq (t1, t2))
| core_term_parser (SMTLIB2.Sym "ite", [t1, t2, t3]) =
let
val T = fastype_of t2
val c = Const (@{const_name HOL.If}, [@{typ HOL.bool}, T, T] ---> T)
in SOME (c $ t1 $ t2 $ t3) end
| core_term_parser (SMTLIB2.Num i, []) = SOME (HOLogic.mk_number @{typ Int.int} i)
| core_term_parser (SMTLIB2.Sym "-", [t]) = SOME (mk_unary @{const_name uminus_class.uminus} t)
| core_term_parser (SMTLIB2.Sym "~", [t]) = SOME (mk_unary @{const_name uminus_class.uminus} t)
| core_term_parser (SMTLIB2.Sym "+", t :: ts) =
SOME (mk_lassoc' @{const_name plus_class.plus} t ts)
| core_term_parser (SMTLIB2.Sym "-", t :: ts) =
SOME (mk_lassoc' @{const_name minus_class.minus} t ts)
| core_term_parser (SMTLIB2.Sym "*", t :: ts) =
SOME (mk_lassoc' @{const_name times_class.times} t ts)
| core_term_parser (SMTLIB2.Sym "div", [t1, t2]) = SOME (mk_binary @{const_name SMT2.z3div} t1 t2)
| core_term_parser (SMTLIB2.Sym "mod", [t1, t2]) = SOME (mk_binary @{const_name SMT2.z3mod} t1 t2)
| core_term_parser (SMTLIB2.Sym "<", [t1, t2]) = SOME (mk_less t1 t2)
| core_term_parser (SMTLIB2.Sym ">", [t1, t2]) = SOME (mk_less t2 t1)
| core_term_parser (SMTLIB2.Sym "<=", [t1, t2]) = SOME (mk_less_eq t1 t2)
| core_term_parser (SMTLIB2.Sym ">=", [t1, t2]) = SOME (mk_less_eq t2 t1)
| core_term_parser _ = NONE
(* custom type and term parsers *)
type type_parser = SMTLIB2.tree * typ list -> typ option
type term_parser = SMTLIB2.tree * term list -> term option
fun id_ord ((id1, _), (id2, _)) = int_ord (id1, id2)
structure Parsers = Generic_Data
(
type T = (int * type_parser) list * (int * term_parser) list
val empty : T = ([(serial (), core_type_parser)], [(serial (), core_term_parser)])
val extend = I
fun merge ((tys1, ts1), (tys2, ts2)) =
(Ord_List.merge id_ord (tys1, tys2), Ord_List.merge id_ord (ts1, ts2))
)
fun add_type_parser type_parser =
Parsers.map (apfst (Ord_List.insert id_ord (serial (), type_parser)))
fun add_term_parser term_parser =
Parsers.map (apsnd (Ord_List.insert id_ord (serial (), term_parser)))
fun get_type_parsers ctxt = map snd (fst (Parsers.get (Context.Proof ctxt)))
fun get_term_parsers ctxt = map snd (snd (Parsers.get (Context.Proof ctxt)))
fun apply_parsers parsers x =
let
fun apply [] = NONE
| apply (parser :: parsers) =
(case parser x of
SOME y => SOME y
| NONE => apply parsers)
in apply parsers end
(* type and term parsing *)
exception SMTLIB2_PARSE of string * SMTLIB2.tree
val desymbolize = Name.desymbolize (SOME false) o perhaps (try (unprefix "?"))
fun fresh_fun add name n T ({ctxt, id, syms, typs, funs, extra}: ('a, 'b) context) =
let
val (n', ctxt') = yield_singleton Variable.variant_fixes n ctxt
val t = Free (n', T)
val funs' = Symtab.update (name, t) funs
in (t, mk_context ctxt' id syms typs funs' (add (n', T) extra)) end
fun declare_fun name = snd oo fresh_fun cons name (desymbolize name)
fun declare_free name = fresh_fun (cons o pair name) name (desymbolize name)
fun parse_type cx ty Ts =
(case apply_parsers (get_type_parsers (ctxt_of cx)) (ty, Ts) of
SOME T => T
| NONE =>
(case ty of
SMTLIB2.Sym name =>
(case lookup_typ cx name of
SOME T => T
| NONE => raise SMTLIB2_PARSE ("unknown SMT type", ty))
| _ => raise SMTLIB2_PARSE ("bad SMT type format", ty)))
fun parse_term t ts cx =
(case apply_parsers (get_term_parsers (ctxt_of cx)) (t, ts) of
SOME u => (u, cx)
| NONE =>
(case t of
SMTLIB2.Sym name =>
(case lookup_fun cx name of
SOME u => (Term.list_comb (u, ts), cx)
| NONE =>
if null ts then declare_free name Term.dummyT cx
else raise SMTLIB2_PARSE ("bad SMT term", t))
| _ => raise SMTLIB2_PARSE ("bad SMT term format", t)))
fun type_of cx ty =
(case try (parse_type cx ty) [] of
SOME T => T
| NONE =>
(case ty of
SMTLIB2.S (ty' :: tys) => parse_type cx ty' (map (type_of cx) tys)
| _ => raise SMTLIB2_PARSE ("bad SMT type", ty)))
fun dest_var cx (SMTLIB2.S [SMTLIB2.Sym name, ty]) = (name, (desymbolize name, type_of cx ty))
| dest_var _ v = raise SMTLIB2_PARSE ("bad SMT quantifier variable format", v)
fun dest_body (SMTLIB2.S (SMTLIB2.Sym "!" :: body :: _)) = dest_body body
| dest_body body = body
fun dest_binding (SMTLIB2.S [SMTLIB2.Sym name, t]) = (name, Tree t)
| dest_binding b = raise SMTLIB2_PARSE ("bad SMT let binding format", b)
fun term_of t cx =
(case t of
SMTLIB2.S [SMTLIB2.Sym "forall", SMTLIB2.S vars, body] => quant HOLogic.mk_all vars body cx
| SMTLIB2.S [SMTLIB2.Sym "exists", SMTLIB2.S vars, body] => quant HOLogic.mk_exists vars body cx
| SMTLIB2.S [SMTLIB2.Sym "let", SMTLIB2.S bindings, body] =>
with_bindings (map dest_binding bindings) (term_of body) cx
| SMTLIB2.S (SMTLIB2.Sym "!" :: t :: _) => term_of t cx
| SMTLIB2.S (f :: args) =>
cx
|> fold_map term_of args
|-> parse_term f
| SMTLIB2.Sym name =>
(case lookup_binding cx name of
Tree u =>
cx
|> term_of u
|-> (fn u' => pair u' o update_binding (name, Term u'))
| Term u => (u, cx)
| None => parse_term t [] cx
| _ => raise SMTLIB2_PARSE ("bad SMT term format", t))
| _ => parse_term t [] cx)
and quant q vars body cx =
let val vs = map (dest_var cx) vars
in
cx
|> with_bindings (map (apsnd (Term o Free)) vs) (term_of (dest_body body))
|>> fold_rev (fn (_, (n, T)) => fn t => q (n, T, t)) vs
end
end;