(*  Title:      HOL/Tools/Sledgehammer/metis_clauses.ML

     Author:     Jia Meng, Cambridge University Computer Laboratory and NICTA

     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 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 {name: string, conclLit: arLit, premLits: arLit list}

   datatype class_rel_clause =

     ClassRelClause of {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

   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 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}, "member"),

                (@{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 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 ("_" ^ stringN_of_int 3 (Char.ord c))  (*fixed width, in case more digits follow*)

 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

 (**** 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 {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, name, (cls,args)) =

   let

     val tvars = gen_TVars (length args)

     val tvars_srts = ListPair.zip (tvars, args)

   in

     ArityClause {name = 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 {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 {name = sub ^ "_" ^ 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

 (*********************************************************************)

 (* 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;