(* Title: HOL/Tools/Sledgehammer/metis_clauses.ML
Author: Jia Meng, Cambridge University Computer Laboratory
Author: Jasmin Blanchette, TU Muenchen
Storing/printing FOL clauses and arity clauses. Typed equality is
treated differently.
*)
signature METIS_CLAUSES =
sig
type name = string * string
datatype kind = Axiom | Conjecture
datatype type_literal =
TyLitVar of name * name |
TyLitFree of name * name
datatype arLit =
TConsLit of name * name * name list |
TVarLit of name * name
datatype arity_clause = ArityClause of
{axiom_name: string, conclLit: arLit, premLits: arLit list}
datatype class_rel_clause = ClassRelClause of
{axiom_name: string, subclass: name, superclass: name}
datatype combtyp =
CombTVar of name |
CombTFree of name |
CombType of name * combtyp list
datatype combterm =
CombConst of name * combtyp * combtyp list (* Const and Free *) |
CombVar of name * combtyp |
CombApp of combterm * combterm
datatype fol_literal = FOLLiteral of bool * combterm
datatype fol_clause =
FOLClause of {clause_id: int, axiom_name: string, th: thm, kind: kind,
literals: fol_literal list, ctypes_sorts: typ list}
val type_wrapper_name : string
val bound_var_prefix : string
val schematic_var_prefix: string
val fixed_var_prefix: string
val tvar_prefix: string
val tfree_prefix: string
val const_prefix: string
val type_const_prefix: string
val class_prefix: string
val union_all: ''a list list -> ''a list
val invert_const: string -> string
val ascii_of: string -> string
val undo_ascii_of: string -> string
val strip_prefix_and_undo_ascii: string -> string -> string option
val make_bound_var : string -> string
val make_schematic_var : string * int -> string
val make_fixed_var : string -> string
val make_schematic_type_var : string * int -> string
val make_fixed_type_var : string -> string
val make_fixed_const : string -> string
val make_fixed_type_const : string -> string
val make_type_class : string -> string
val skolem_theory_name: string
val skolem_prefix: string
val skolem_infix: string
val is_skolem_const_name: string -> bool
val num_type_args: theory -> string -> int
val type_literals_for_types : typ list -> type_literal list
val make_class_rel_clauses: theory -> class list -> class list -> class_rel_clause list
val make_arity_clauses: theory -> string list -> class list -> class list * arity_clause list
val combtyp_of : combterm -> combtyp
val strip_combterm_comb : combterm -> combterm * combterm list
val combterm_from_term :
theory -> (string * typ) list -> term -> combterm * typ list
val literals_of_term : theory -> term -> fol_literal list * typ list
val conceal_skolem_terms :
int -> (string * term) list -> term -> (string * term) list * term
val reveal_skolem_terms : (string * term) list -> term -> term
val tfree_classes_of_terms : term list -> string list
val tvar_classes_of_terms : term list -> string list
val type_consts_of_terms : theory -> term list -> string list
end
structure Metis_Clauses : METIS_CLAUSES =
struct
val type_wrapper_name = "ti"
val bound_var_prefix = "B_"
val schematic_var_prefix = "V_"
val fixed_var_prefix = "v_"
val tvar_prefix = "T_";
val tfree_prefix = "t_";
val class_rel_clause_prefix = "clsrel_";
val const_prefix = "c_";
val type_const_prefix = "tc_";
val class_prefix = "class_";
fun union_all xss = fold (union (op =)) xss []
(* Readable names for the more common symbolic functions. Do not mess with the
last nine entries of the table unless you know what you are doing. *)
val const_trans_table =
Symtab.make [(@{type_name Product_Type.prod}, "prod"),
(@{type_name Sum_Type.sum}, "sum"),
(@{const_name "op ="}, "equal"),
(@{const_name "op &"}, "and"),
(@{const_name "op |"}, "or"),
(@{const_name "op -->"}, "implies"),
(@{const_name Set.member}, "in"),
(@{const_name fequal}, "fequal"),
(@{const_name COMBI}, "COMBI"),
(@{const_name COMBK}, "COMBK"),
(@{const_name COMBB}, "COMBB"),
(@{const_name COMBC}, "COMBC"),
(@{const_name COMBS}, "COMBS"),
(@{const_name True}, "True"),
(@{const_name False}, "False"),
(@{const_name If}, "If")]
(* Invert the table of translations between Isabelle and ATPs. *)
val const_trans_table_inv =
Symtab.update ("fequal", @{const_name "op ="})
(Symtab.make (map swap (Symtab.dest const_trans_table)))
val invert_const = perhaps (Symtab.lookup const_trans_table_inv)
(*Escaping of special characters.
Alphanumeric characters are left unchanged.
The character _ goes to __
Characters in the range ASCII space to / go to _A to _P, respectively.
Other printing characters go to _nnn where nnn is the decimal ASCII code.*)
val A_minus_space = Char.ord #"A" - Char.ord #" ";
fun stringN_of_int 0 _ = ""
| stringN_of_int k n = stringN_of_int (k-1) (n div 10) ^ Int.toString (n mod 10);
fun ascii_of_c c =
if Char.isAlphaNum c then String.str c
else if c = #"_" then "__"
else if #" " <= c andalso c <= #"/"
then "_" ^ String.str (Char.chr (Char.ord c + A_minus_space))
else if Char.isPrint c
then ("_" ^ stringN_of_int 3 (Char.ord c)) (*fixed width, in case more digits follow*)
else ""
val ascii_of = String.translate ascii_of_c;
(** Remove ASCII armouring from names in proof files **)
(*We don't raise error exceptions because this code can run inside the watcher.
Also, the errors are "impossible" (hah!)*)
fun undo_ascii_aux rcs [] = String.implode(rev rcs)
| undo_ascii_aux rcs [#"_"] = undo_ascii_aux (#"_"::rcs) [] (*ERROR*)
(*Three types of _ escapes: __, _A to _P, _nnn*)
| undo_ascii_aux rcs (#"_" :: #"_" :: cs) = undo_ascii_aux (#"_"::rcs) cs
| undo_ascii_aux rcs (#"_" :: c :: cs) =
if #"A" <= c andalso c<= #"P" (*translation of #" " to #"/"*)
then undo_ascii_aux (Char.chr(Char.ord c - A_minus_space) :: rcs) cs
else
let val digits = List.take (c::cs, 3) handle Subscript => []
in
case Int.fromString (String.implode digits) of
NONE => undo_ascii_aux (c:: #"_"::rcs) cs (*ERROR*)
| SOME n => undo_ascii_aux (Char.chr n :: rcs) (List.drop (cs, 2))
end
| undo_ascii_aux rcs (c::cs) = undo_ascii_aux (c::rcs) cs;
val undo_ascii_of = undo_ascii_aux [] o String.explode;
(* If string s has the prefix s1, return the result of deleting it,
un-ASCII'd. *)
fun strip_prefix_and_undo_ascii s1 s =
if String.isPrefix s1 s then
SOME (undo_ascii_of (String.extract (s, size s1, NONE)))
else
NONE
(*Remove the initial ' character from a type variable, if it is present*)
fun trim_type_var s =
if s <> "" andalso String.sub(s,0) = #"'" then String.extract(s,1,NONE)
else error ("trim_type: Malformed type variable encountered: " ^ s);
fun ascii_of_indexname (v,0) = ascii_of v
| ascii_of_indexname (v,i) = ascii_of v ^ "_" ^ Int.toString i;
fun make_bound_var x = bound_var_prefix ^ ascii_of x
fun make_schematic_var v = schematic_var_prefix ^ ascii_of_indexname v
fun make_fixed_var x = fixed_var_prefix ^ ascii_of x
fun make_schematic_type_var (x,i) =
tvar_prefix ^ (ascii_of_indexname (trim_type_var x,i));
fun make_fixed_type_var x = tfree_prefix ^ (ascii_of (trim_type_var x));
fun lookup_const c =
case Symtab.lookup const_trans_table c of
SOME c' => c'
| NONE => ascii_of c
(* "op =" MUST BE "equal" because it's built into ATPs. *)
fun make_fixed_const @{const_name "op ="} = "equal"
| make_fixed_const c = const_prefix ^ lookup_const c
fun make_fixed_type_const c = type_const_prefix ^ lookup_const c
fun make_type_class clas = class_prefix ^ ascii_of clas;
val skolem_theory_name = "Sledgehammer" ^ Long_Name.separator ^ "Sko"
val skolem_prefix = "sko_"
val skolem_infix = "$"
(* Hack: Could return false positives (e.g., a user happens to declare a
constant called "SomeTheory.sko_means_shoe_in_$wedish". *)
val is_skolem_const_name =
Long_Name.base_name
#> String.isPrefix skolem_prefix andf String.isSubstring skolem_infix
(* The number of type arguments of a constant, zero if it's monomorphic. For
(instances of) Skolem pseudoconstants, this information is encoded in the
constant name. *)
fun num_type_args thy s =
if String.isPrefix skolem_theory_name s then
s |> unprefix skolem_theory_name
|> space_explode skolem_infix |> hd
|> space_explode "_" |> List.last |> Int.fromString |> the
else
(s, Sign.the_const_type thy s) |> Sign.const_typargs thy |> length
(**** Definitions and functions for FOL clauses for TPTP format output ****)
type name = string * string
datatype kind = Axiom | Conjecture
(**** Isabelle FOL clauses ****)
(* The first component is the type class; the second is a TVar or TFree. *)
datatype type_literal =
TyLitVar of name * name |
TyLitFree of name * name
exception CLAUSE of string * term;
(*Make literals for sorted type variables*)
fun sorts_on_typs_aux (_, []) = []
| sorts_on_typs_aux ((x,i), s::ss) =
let val sorts = sorts_on_typs_aux ((x,i), ss)
in
if s = "HOL.type" then sorts
else if i = ~1 then TyLitFree (`make_type_class s, `make_fixed_type_var x) :: sorts
else TyLitVar (`make_type_class s, (make_schematic_type_var (x,i), x)) :: sorts
end;
fun sorts_on_typs (TFree (a,s)) = sorts_on_typs_aux ((a,~1),s)
| sorts_on_typs (TVar (v,s)) = sorts_on_typs_aux (v,s);
(*Given a list of sorted type variables, return a list of type literals.*)
fun type_literals_for_types Ts =
fold (union (op =)) (map sorts_on_typs Ts) []
(** make axiom and conjecture clauses. **)
(**** Isabelle arities ****)
datatype arLit =
TConsLit of name * name * name list |
TVarLit of name * name
datatype arity_clause =
ArityClause of {axiom_name: string, conclLit: arLit, premLits: arLit list}
fun gen_TVars 0 = []
| gen_TVars n = ("T_" ^ Int.toString n) :: gen_TVars (n-1);
fun pack_sort(_,[]) = []
| pack_sort(tvar, "HOL.type"::srt) = pack_sort (tvar, srt) (*IGNORE sort "type"*)
| pack_sort(tvar, cls::srt) =
(`make_type_class cls, (tvar, tvar)) :: pack_sort (tvar, srt)
(*Arity of type constructor tcon :: (arg1,...,argN)res*)
fun make_axiom_arity_clause (tcons, axiom_name, (cls,args)) =
let
val tvars = gen_TVars (length args)
val tvars_srts = ListPair.zip (tvars, args)
in
ArityClause {axiom_name = axiom_name,
conclLit = TConsLit (`make_type_class cls,
`make_fixed_type_const tcons,
tvars ~~ tvars),
premLits = map TVarLit (union_all (map pack_sort tvars_srts))}
end
(**** Isabelle class relations ****)
datatype class_rel_clause =
ClassRelClause of {axiom_name: string, subclass: name, superclass: name}
(*Generate all pairs (sub,super) such that sub is a proper subclass of super in theory thy.*)
fun class_pairs _ [] _ = []
| class_pairs thy subs supers =
let
val class_less = Sorts.class_less (Sign.classes_of thy)
fun add_super sub super = class_less (sub, super) ? cons (sub, super)
fun add_supers sub = fold (add_super sub) supers
in fold add_supers subs [] end
fun make_class_rel_clause (sub,super) =
ClassRelClause {axiom_name = class_rel_clause_prefix ^ ascii_of sub ^ "_" ^
ascii_of super,
subclass = `make_type_class sub,
superclass = `make_type_class super};
fun make_class_rel_clauses thy subs supers =
map make_class_rel_clause (class_pairs thy subs supers);
(** Isabelle arities **)
fun arity_clause _ _ (_, []) = []
| arity_clause seen n (tcons, ("HOL.type",_)::ars) = (*ignore*)
arity_clause seen n (tcons,ars)
| arity_clause seen n (tcons, (ar as (class,_)) :: ars) =
if member (op =) seen class then (*multiple arities for the same tycon, class pair*)
make_axiom_arity_clause (tcons, lookup_const tcons ^ "_" ^ class ^ "_" ^ Int.toString n, ar) ::
arity_clause seen (n+1) (tcons,ars)
else
make_axiom_arity_clause (tcons, lookup_const tcons ^ "_" ^ class, ar) ::
arity_clause (class::seen) n (tcons,ars)
fun multi_arity_clause [] = []
| multi_arity_clause ((tcons, ars) :: tc_arlists) =
arity_clause [] 1 (tcons, ars) @ multi_arity_clause tc_arlists
(*Generate all pairs (tycon,class,sorts) such that tycon belongs to class in theory thy
provided its arguments have the corresponding sorts.*)
fun type_class_pairs thy tycons classes =
let val alg = Sign.classes_of thy
fun domain_sorts tycon = Sorts.mg_domain alg tycon o single
fun add_class tycon class =
cons (class, domain_sorts tycon class)
handle Sorts.CLASS_ERROR _ => I
fun try_classes tycon = (tycon, fold (add_class tycon) classes [])
in map try_classes tycons end;
(*Proving one (tycon, class) membership may require proving others, so iterate.*)
fun iter_type_class_pairs _ _ [] = ([], [])
| iter_type_class_pairs thy tycons classes =
let val cpairs = type_class_pairs thy tycons classes
val newclasses = union_all (union_all (union_all (map (map #2 o #2) cpairs)))
|> subtract (op =) classes |> subtract (op =) HOLogic.typeS
val (classes', cpairs') = iter_type_class_pairs thy tycons newclasses
in (union (op =) classes' classes, union (op =) cpairs' cpairs) end;
fun make_arity_clauses thy tycons classes =
let val (classes', cpairs) = iter_type_class_pairs thy tycons classes
in (classes', multi_arity_clause cpairs) end;
datatype combtyp =
CombTVar of name |
CombTFree of name |
CombType of name * combtyp list
datatype combterm =
CombConst of name * combtyp * combtyp list (* Const and Free *) |
CombVar of name * combtyp |
CombApp of combterm * combterm
datatype fol_literal = FOLLiteral of bool * combterm
datatype fol_clause =
FOLClause of {clause_id: int, axiom_name: string, th: thm, kind: kind,
literals: fol_literal list, ctypes_sorts: typ list}
(*********************************************************************)
(* convert a clause with type Term.term to a clause with type clause *)
(*********************************************************************)
(*Result of a function type; no need to check that the argument type matches.*)
fun result_type (CombType (_, [_, tp2])) = tp2
| result_type _ = raise Fail "non-function type"
fun combtyp_of (CombConst (_, tp, _)) = tp
| combtyp_of (CombVar (_, tp)) = tp
| combtyp_of (CombApp (t1, _)) = result_type (combtyp_of t1)
(*gets the head of a combinator application, along with the list of arguments*)
fun strip_combterm_comb u =
let fun stripc (CombApp(t,u), ts) = stripc (t, u::ts)
| stripc x = x
in stripc(u,[]) end
fun type_of (Type (a, Ts)) =
let val (folTypes,ts) = types_of Ts in
(CombType (`make_fixed_type_const a, folTypes), ts)
end
| type_of (tp as TFree (a, _)) = (CombTFree (`make_fixed_type_var a), [tp])
| type_of (tp as TVar (x, _)) =
(CombTVar (make_schematic_type_var x, string_of_indexname x), [tp])
and types_of Ts =
let val (folTyps, ts) = ListPair.unzip (map type_of Ts) in
(folTyps, union_all ts)
end
(* same as above, but no gathering of sort information *)
fun simp_type_of (Type (a, Ts)) =
CombType (`make_fixed_type_const a, map simp_type_of Ts)
| simp_type_of (TFree (a, _)) = CombTFree (`make_fixed_type_var a)
| simp_type_of (TVar (x, _)) =
CombTVar (make_schematic_type_var x, string_of_indexname x)
(* Converts a term (with combinators) into a combterm. Also accummulates sort
infomation. *)
fun combterm_from_term thy bs (P $ Q) =
let val (P', tsP) = combterm_from_term thy bs P
val (Q', tsQ) = combterm_from_term thy bs Q
in (CombApp (P', Q'), union (op =) tsP tsQ) end
| combterm_from_term thy _ (Const (c, T)) =
let
val (tp, ts) = type_of T
val tvar_list =
(if String.isPrefix skolem_theory_name c then
[] |> Term.add_tvarsT T |> map TVar
else
(c, T) |> Sign.const_typargs thy)
|> map simp_type_of
val c' = CombConst (`make_fixed_const c, tp, tvar_list)
in (c',ts) end
| combterm_from_term _ _ (Free (v, T)) =
let val (tp,ts) = type_of T
val v' = CombConst (`make_fixed_var v, tp, [])
in (v',ts) end
| combterm_from_term _ _ (Var (v, T)) =
let val (tp,ts) = type_of T
val v' = CombVar ((make_schematic_var v, string_of_indexname v), tp)
in (v',ts) end
| combterm_from_term _ bs (Bound j) =
let
val (s, T) = nth bs j
val (tp, ts) = type_of T
val v' = CombConst (`make_bound_var s, tp, [])
in (v', ts) end
| combterm_from_term _ _ (Abs _) = raise Fail "HOL clause: Abs"
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 "op |"} $ 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
(FOLLiteral (pol, pred) :: lits, union (op =) ts ts')
end
val literals_of_term = literals_of_term1 ([], [])
fun skolem_name i j num_T_args =
skolem_prefix ^ (space_implode "_" (map Int.toString [i, j, num_T_args])) ^
skolem_infix ^ "g"
fun conceal_skolem_terms i skolems t =
if exists_Const (curry (op =) @{const_name skolem_id} o fst) t then
let
fun aux skolems
(t as (Const (@{const_name skolem_id}, Type (_, [_, T])) $ _)) =
let
val (skolems, s) =
if i = ~1 then
(skolems, @{const_name undefined})
else case AList.find (op aconv) skolems t of
s :: _ => (skolems, s)
| [] =>
let
val s = skolem_theory_name ^ "." ^
skolem_name i (length skolems)
(length (Term.add_tvarsT T []))
in ((s, t) :: skolems, s) end
in (skolems, Const (s, T)) end
| aux skolems (t1 $ t2) =
let
val (skolems, t1) = aux skolems t1
val (skolems, t2) = aux skolems t2
in (skolems, t1 $ t2) end
| aux skolems (Abs (s, T, t')) =
let val (skolems, t') = aux skolems t' in
(skolems, Abs (s, T, t'))
end
| aux skolems t = (skolems, t)
in aux skolems t end
else
(skolems, t)
fun reveal_skolem_terms skolems =
map_aterms (fn t as Const (s, _) =>
if String.isPrefix skolem_theory_name s then
AList.lookup (op =) skolems s |> the
|> map_types Type_Infer.paramify_vars
else
t
| t => t)
(***************************************************************)
(* Type Classes Present in the Axiom or Conjecture Clauses *)
(***************************************************************)
fun set_insert (x, s) = Symtab.update (x, ()) s
fun add_classes (sorts, cset) = List.foldl set_insert cset (flat sorts)
(*Remove this trivial type class*)
fun delete_type cset = Symtab.delete_safe (the_single @{sort HOL.type}) cset;
fun tfree_classes_of_terms ts =
let val sorts_list = map (map #2 o OldTerm.term_tfrees) ts
in Symtab.keys (delete_type (List.foldl add_classes Symtab.empty sorts_list)) end;
fun tvar_classes_of_terms ts =
let val sorts_list = map (map #2 o OldTerm.term_tvars) ts
in Symtab.keys (delete_type (List.foldl add_classes Symtab.empty sorts_list)) end;
(*fold type constructors*)
fun fold_type_consts f (Type (a, Ts)) x = fold (fold_type_consts f) Ts (f (a,x))
| fold_type_consts _ _ x = x;
(*Type constructors used to instantiate overloaded constants are the only ones needed.*)
fun add_type_consts_in_term thy =
let
val const_typargs = Sign.const_typargs thy
fun aux (Const x) = fold (fold_type_consts set_insert) (const_typargs x)
| aux (Abs (_, _, u)) = aux u
| aux (Const (@{const_name skolem_id}, _) $ _) = I
| aux (t $ u) = aux t #> aux u
| aux _ = I
in aux end
fun type_consts_of_terms thy ts =
Symtab.keys (fold (add_type_consts_in_term thy) ts Symtab.empty);
end;