(* Title: HOL/Tools/Metis/metis_translate.ML
Author: Jia Meng, Cambridge University Computer Laboratory and NICTA
Author: Kong W. Susanto, Cambridge University Computer Laboratory
Author: Lawrence C. Paulson, Cambridge University Computer Laboratory
Author: Jasmin Blanchette, TU Muenchen
Translation of HOL to FOL for Metis.
*)
signature METIS_TRANSLATE =
sig
type type_literal = ATP_Translate.type_literal
type type_sys = ATP_Translate.type_sys
datatype mode = FO | HO | FT | New
type metis_problem =
{axioms : (Metis_Thm.thm * thm) list,
tfrees : type_literal list,
old_skolems : (string * term) list}
val metis_equal : string
val metis_predicator : string
val metis_app_op : string
val metis_generated_var_prefix : string
val metis_name_table : ((string * int) * string) list
val reveal_old_skolem_terms : (string * term) list -> term -> term
val string_of_mode : mode -> string
val prepare_metis_problem :
Proof.context -> mode -> type_sys option -> thm list -> thm list
-> mode * int Symtab.table * metis_problem
end
structure Metis_Translate : METIS_TRANSLATE =
struct
open ATP_Problem
open ATP_Translate
val metis_equal = "="
val metis_predicator = "{}"
val metis_app_op = "."
val metis_generated_var_prefix = "_"
val metis_name_table =
[((tptp_equal, 2), metis_equal),
((tptp_old_equal, 2), metis_equal),
((const_prefix ^ predicator_name, 1), metis_predicator),
((const_prefix ^ app_op_name, 2), metis_app_op)]
fun predicate_of thy ((@{const Not} $ P), pos) = predicate_of thy (P, not pos)
| predicate_of thy (t, pos) =
(combterm_from_term thy [] (Envir.eta_contract t), pos)
fun literals_of_term1 args thy (@{const Trueprop} $ P) =
literals_of_term1 args thy P
| literals_of_term1 args thy (@{const HOL.disj} $ P $ Q) =
literals_of_term1 (literals_of_term1 args thy P) thy Q
| literals_of_term1 (lits, ts) thy P =
let val ((pred, ts'), pol) = predicate_of thy (P, true) in
((pol, pred) :: lits, union (op =) ts ts')
end
val literals_of_term = literals_of_term1 ([], [])
fun old_skolem_const_name i j num_T_args =
old_skolem_const_prefix ^ Long_Name.separator ^
(space_implode Long_Name.separator (map string_of_int [i, j, num_T_args]))
fun conceal_old_skolem_terms i old_skolems t =
if exists_Const (curry (op =) @{const_name Meson.skolem} o fst) t then
let
fun aux old_skolems
(t as (Const (@{const_name Meson.skolem}, Type (_, [_, T])) $ _)) =
let
val (old_skolems, s) =
if i = ~1 then
(old_skolems, @{const_name undefined})
else case AList.find (op aconv) old_skolems t of
s :: _ => (old_skolems, s)
| [] =>
let
val s = old_skolem_const_name i (length old_skolems)
(length (Term.add_tvarsT T []))
in ((s, t) :: old_skolems, s) end
in (old_skolems, Const (s, T)) end
| aux old_skolems (t1 $ t2) =
let
val (old_skolems, t1) = aux old_skolems t1
val (old_skolems, t2) = aux old_skolems t2
in (old_skolems, t1 $ t2) end
| aux old_skolems (Abs (s, T, t')) =
let val (old_skolems, t') = aux old_skolems t' in
(old_skolems, Abs (s, T, t'))
end
| aux old_skolems t = (old_skolems, t)
in aux old_skolems t end
else
(old_skolems, t)
fun reveal_old_skolem_terms old_skolems =
map_aterms (fn t as Const (s, _) =>
if String.isPrefix old_skolem_const_prefix s then
AList.lookup (op =) old_skolems s |> the
|> map_types Type_Infer.paramify_vars
else
t
| t => t)
(* ------------------------------------------------------------------------- *)
(* HOL to FOL (Isabelle to Metis) *)
(* ------------------------------------------------------------------------- *)
(* first-order, higher-order, fully-typed, new *)
datatype mode = FO | HO | FT | New
fun string_of_mode FO = "FO"
| string_of_mode HO = "HO"
| string_of_mode FT = "FT"
| string_of_mode New = "New"
fun fn_isa_to_met_sublevel "equal" = "c_fequal"
| fn_isa_to_met_sublevel "c_False" = "c_fFalse"
| fn_isa_to_met_sublevel "c_True" = "c_fTrue"
| fn_isa_to_met_sublevel "c_Not" = "c_fNot"
| fn_isa_to_met_sublevel "c_conj" = "c_fconj"
| fn_isa_to_met_sublevel "c_disj" = "c_fdisj"
| fn_isa_to_met_sublevel "c_implies" = "c_fimplies"
| fn_isa_to_met_sublevel x = x
fun fn_isa_to_met_toplevel "equal" = metis_equal
| fn_isa_to_met_toplevel x = x
fun metis_lit b c args = (b, (c, args));
fun metis_term_from_typ (Type (s, Ts)) =
Metis_Term.Fn (make_fixed_type_const s, map metis_term_from_typ Ts)
| metis_term_from_typ (TFree (s, _)) =
Metis_Term.Fn (make_fixed_type_var s, [])
| metis_term_from_typ (TVar (x, _)) =
Metis_Term.Var (make_schematic_type_var x)
(*These two functions insert type literals before the real literals. That is the
opposite order from TPTP linkup, but maybe OK.*)
fun hol_term_to_fol_FO tm =
case strip_combterm_comb tm of
(CombConst ((c, _), _, Ts), tms) =>
let val tyargs = map metis_term_from_typ Ts
val args = map hol_term_to_fol_FO tms
in Metis_Term.Fn (c, tyargs @ args) end
| (CombVar ((v, _), _), []) => Metis_Term.Var v
| _ => raise Fail "non-first-order combterm"
fun hol_term_to_fol_HO (CombConst ((a, _), _, Ts)) =
Metis_Term.Fn (fn_isa_to_met_sublevel a, map metis_term_from_typ Ts)
| hol_term_to_fol_HO (CombVar ((s, _), _)) = Metis_Term.Var s
| hol_term_to_fol_HO (CombApp (tm1, tm2)) =
Metis_Term.Fn (metis_app_op, map hol_term_to_fol_HO [tm1, tm2])
(*The fully-typed translation, to avoid type errors*)
fun tag_with_type tm T =
Metis_Term.Fn (type_tag_name, [tm, metis_term_from_typ T])
fun hol_term_to_fol_FT (CombVar ((s, _), ty)) =
tag_with_type (Metis_Term.Var s) ty
| hol_term_to_fol_FT (CombConst ((a, _), ty, _)) =
tag_with_type (Metis_Term.Fn (fn_isa_to_met_sublevel a, [])) ty
| hol_term_to_fol_FT (tm as CombApp (tm1,tm2)) =
tag_with_type
(Metis_Term.Fn (metis_app_op, map hol_term_to_fol_FT [tm1, tm2]))
(combtyp_of tm)
fun hol_literal_to_fol FO (pos, tm) =
let
val (CombConst((p, _), _, Ts), tms) = strip_combterm_comb tm
val tylits = if p = "equal" then [] else map metis_term_from_typ Ts
val lits = map hol_term_to_fol_FO tms
in metis_lit pos (fn_isa_to_met_toplevel p) (tylits @ lits) end
| hol_literal_to_fol HO (pos, tm) =
(case strip_combterm_comb tm of
(CombConst(("equal", _), _, _), tms) =>
metis_lit pos metis_equal (map hol_term_to_fol_HO tms)
| _ => metis_lit pos metis_predicator [hol_term_to_fol_HO tm])
| hol_literal_to_fol FT (pos, tm) =
(case strip_combterm_comb tm of
(CombConst(("equal", _), _, _), tms) =>
metis_lit pos metis_equal (map hol_term_to_fol_FT tms)
| _ => metis_lit pos metis_predicator [hol_term_to_fol_FT tm])
fun literals_of_hol_term thy mode t =
let val (lits, types_sorts) = literals_of_term thy t in
(map (hol_literal_to_fol mode) lits, types_sorts)
end
(*Sign should be "true" for conjecture type constraints, "false" for type lits in clauses.*)
fun metis_of_type_literals pos (TyLitVar ((s, _), (s', _))) =
metis_lit pos s [Metis_Term.Var s']
| metis_of_type_literals pos (TyLitFree ((s, _), (s', _))) =
metis_lit pos s [Metis_Term.Fn (s',[])]
fun has_default_sort _ (TVar _) = false
| has_default_sort ctxt (TFree (x, s)) =
(s = the_default [] (Variable.def_sort ctxt (x, ~1)));
fun metis_of_tfree tf =
Metis_Thm.axiom (Metis_LiteralSet.singleton (metis_of_type_literals true tf));
fun hol_thm_to_fol is_conjecture ctxt mode j old_skolems th =
let
val thy = Proof_Context.theory_of ctxt
val (old_skolems, (mlits, types_sorts)) =
th |> prop_of |> Logic.strip_imp_concl
|> conceal_old_skolem_terms j old_skolems
||> (HOLogic.dest_Trueprop #> literals_of_hol_term thy mode)
in
if is_conjecture then
(Metis_Thm.axiom (Metis_LiteralSet.fromList mlits),
raw_type_literals_for_types types_sorts, old_skolems)
else
let
val tylits = types_sorts |> filter_out (has_default_sort ctxt)
|> raw_type_literals_for_types
val mtylits = map (metis_of_type_literals false) tylits
in
(Metis_Thm.axiom (Metis_LiteralSet.fromList(mtylits @ mlits)), [],
old_skolems)
end
end;
(* ------------------------------------------------------------------------- *)
(* Logic maps manage the interface between HOL and first-order logic. *)
(* ------------------------------------------------------------------------- *)
type metis_problem =
{axioms : (Metis_Thm.thm * thm) list,
tfrees : type_literal list,
old_skolems : (string * term) list}
fun is_quasi_fol_clause thy =
Meson.is_fol_term thy o snd o conceal_old_skolem_terms ~1 [] o prop_of
(*Extract TFree constraints from context to include as conjecture clauses*)
fun init_tfrees ctxt =
let fun add ((a,i),s) Ts = if i = ~1 then TFree(a,s) :: Ts else Ts in
Vartab.fold add (#2 (Variable.constraints_of ctxt)) []
|> raw_type_literals_for_types
end;
fun const_in_metis c (pred, tm_list) =
let
fun in_mterm (Metis_Term.Var _) = false
| in_mterm (Metis_Term.Fn (nm, tm_list)) =
c = nm orelse exists in_mterm tm_list
in c = pred orelse exists in_mterm tm_list end
(* ARITY CLAUSE *)
fun m_arity_cls (TConsLit ((c, _), (t, _), args)) =
metis_lit true c [Metis_Term.Fn(t, map (Metis_Term.Var o fst) args)]
| m_arity_cls (TVarLit ((c, _), (s, _))) =
metis_lit false c [Metis_Term.Var s]
(*TrueI is returned as the Isabelle counterpart because there isn't any.*)
fun arity_cls ({prem_lits, concl_lits, ...} : arity_clause) =
(TrueI,
Metis_Thm.axiom (Metis_LiteralSet.fromList
(map m_arity_cls (concl_lits :: prem_lits))));
(* CLASSREL CLAUSE *)
fun m_class_rel_cls (subclass, _) (superclass, _) =
[metis_lit false subclass [Metis_Term.Var "T"],
metis_lit true superclass [Metis_Term.Var "T"]]
fun class_rel_cls ({subclass, superclass, ...} : class_rel_clause) =
(TrueI, m_class_rel_cls subclass superclass
|> Metis_LiteralSet.fromList |> Metis_Thm.axiom)
fun type_ext thy tms =
let
val subs = tfree_classes_of_terms tms
val supers = tvar_classes_of_terms tms
val tycons = type_consts_of_terms thy tms
val (supers', arity_clauses) = make_arity_clauses thy tycons supers
val class_rel_clauses = make_class_rel_clauses thy subs supers'
in map class_rel_cls class_rel_clauses @ map arity_cls arity_clauses end
fun metis_name_from_atp s ary =
AList.lookup (op =) metis_name_table (s, ary) |> the_default s
fun metis_term_from_atp (ATerm (s, tms)) =
if is_tptp_variable s then
Metis_Term.Var s
else
Metis_Term.Fn (metis_name_from_atp s (length tms),
map metis_term_from_atp tms)
fun metis_atom_from_atp (AAtom (ATerm (s, tms))) =
(metis_name_from_atp s (length tms), map metis_term_from_atp tms)
| metis_atom_from_atp _ = raise Fail "not CNF -- expected atom"
fun metis_literal_from_atp (AConn (ANot, [phi])) =
(false, metis_atom_from_atp phi)
| metis_literal_from_atp phi = (true, metis_atom_from_atp phi)
fun metis_literals_from_atp (AConn (AOr, [phi1, phi2])) =
uncurry (union (op =)) (pairself metis_literals_from_atp (phi1, phi2))
| metis_literals_from_atp phi = [metis_literal_from_atp phi]
fun metis_axiom_from_atp clauses (Formula (ident, _, phi, _, _)) =
(phi |> metis_literals_from_atp |> Metis_LiteralSet.fromList
|> Metis_Thm.axiom,
case try (unprefix conjecture_prefix) ident of
SOME s => Meson.make_meta_clause (nth clauses (the (Int.fromString s)))
| NONE => TrueI)
| metis_axiom_from_atp _ _ = raise Fail "not CNF -- expected formula"
val default_type_sys = Preds (Polymorphic, Nonmonotonic_Types, Light)
(* Function to generate metis clauses, including comb and type clauses *)
fun prepare_metis_problem ctxt New type_sys conj_clauses fact_clauses =
let
val type_sys = type_sys |> the_default default_type_sys
val explicit_apply = NONE
val clauses = conj_clauses @ fact_clauses
val (atp_problem, _, _, _, _, _, sym_tab) =
prepare_atp_problem ctxt CNF Hypothesis Axiom type_sys explicit_apply
false false (map prop_of clauses) @{prop False} []
val axioms =
atp_problem
|> maps (map_filter (try (metis_axiom_from_atp clauses)) o snd)
in
(New, sym_tab,
{axioms = axioms, tfrees = [], old_skolems = [] (* FIXME ### *)})
end
| prepare_metis_problem ctxt mode _ conj_clauses fact_clauses =
let
val thy = Proof_Context.theory_of ctxt
(* The modes FO and FT are sticky. HO can be downgraded to FO. *)
val mode =
if mode = HO andalso
forall (forall (is_quasi_fol_clause thy))
[conj_clauses, fact_clauses] then
FO
else
mode
fun add_thm is_conjecture (isa_ith, metis_ith)
{axioms, tfrees, old_skolems} : metis_problem =
let
val (mth, tfree_lits, old_skolems) =
hol_thm_to_fol is_conjecture ctxt mode (length axioms) old_skolems
metis_ith
in
{axioms = (mth, isa_ith) :: axioms,
tfrees = union (op =) tfree_lits tfrees, old_skolems = old_skolems}
end;
fun add_type_thm (ith, mth) {axioms, tfrees, old_skolems} =
{axioms = (mth, ith) :: axioms, tfrees = tfrees,
old_skolems = old_skolems}
fun add_tfrees {axioms, tfrees, old_skolems} =
{axioms =
map (rpair TrueI o metis_of_tfree) (distinct (op =) tfrees) @ axioms,
tfrees = tfrees, old_skolems = old_skolems}
val problem =
{axioms = [], tfrees = init_tfrees ctxt, old_skolems = []}
|> fold (add_thm true o `Meson.make_meta_clause) conj_clauses
|> add_tfrees
|> fold (add_thm false o `Meson.make_meta_clause) fact_clauses
val clause_lists = map (Metis_Thm.clause o #1) (#axioms problem)
fun is_used c =
exists (Metis_LiteralSet.exists (const_in_metis c o #2)) clause_lists
val problem =
if mode = FO then
problem
else
let
val fdefs = @{thms fFalse_def fTrue_def fNot_def fconj_def fdisj_def
fimplies_def fequal_def}
val prepare_helper =
zero_var_indexes
#> `(Meson.make_meta_clause
#> rewrite_rule (map safe_mk_meta_eq fdefs))
val helper_ths =
helper_table
|> filter (is_used o prefix const_prefix o fst)
|> maps (fn (_, (needs_full_types, thms)) =>
if needs_full_types andalso mode <> FT then []
else map prepare_helper thms)
in problem |> fold (add_thm false) helper_ths end
val type_ths = type_ext thy (map prop_of (conj_clauses @ fact_clauses))
in (mode, Symtab.empty, fold add_type_thm type_ths problem) end
end;