(*  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 string * name |

    TyLitFree of string * name

  datatype arLit =

      TConsLit of class * string * string list

    | TVarLit of class * string

  datatype arity_clause = ArityClause of

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

  datatype classrel_clause = ClassrelClause of

   {axiom_name: string, subclass: class, superclass: class}

  datatype combtyp =

    TyVar of name |

    TyFree of name |

    TyConstr of name * combtyp list

  datatype combterm =

    CombConst of name * combtyp * combtyp list (* Const and Free *) |

    CombVar of name * combtyp |

    CombApp of combterm * combterm

  datatype literal = Literal of bool * combterm

  datatype hol_clause =

    HOLClause of {clause_id: int, axiom_name: string, th: thm, kind: kind,

                  literals: literal list, ctypes_sorts: typ list}

  val type_wrapper_name : string

  val schematic_var_prefix: string

  val fixed_var_prefix: string

  val tvar_prefix: string

  val tfree_prefix: string

  val const_prefix: string

  val tconst_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: string -> string -> string option

  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_classrel_clauses: theory -> class list -> class list -> classrel_clause list

  val make_arity_clauses: theory -> string list -> class list -> class list * arity_clause list

  val type_of_combterm : combterm -> combtyp

  val strip_combterm_comb : combterm -> combterm * combterm list

  val literals_of_term : theory -> term -> 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

open Clausifier

val type_wrapper_name = "ti"

val schematic_var_prefix = "V_";

val fixed_var_prefix = "v_";

val tvar_prefix = "T_";

val tfree_prefix = "t_";

val classrel_clause_prefix = "clsrel_";

val const_prefix = "c_";

val tconst_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 [(@{const_name "op ="}, "equal"),

               (@{const_name "op &"}, "and"),

               (@{const_name "op |"}, "or"),

               (@{const_name "op -->"}, "implies"),

               (@{const_name "op :"}, "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"),

               (@{type_name "*"}, "prod"),

               (@{type_name "+"}, "sum")]

(* 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 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_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 = tconst_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 string * name |

  TyLitFree of string * 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 class * string * string list

               | TVarLit of class * string;

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) =  (cls, 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 (cls, make_fixed_type_const tcons, tvars),

                  premLits = map TVarLit (union_all(map pack_sort tvars_srts))}

   end;

(**** Isabelle class relations ****)

datatype classrel_clause =

  ClassrelClause of {axiom_name: string, subclass: class, superclass: class}

(*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_classrel_clause (sub,super) =

  ClassrelClause {axiom_name = classrel_clause_prefix ^ ascii_of sub ^ "_" ^

                               ascii_of super,

                  subclass = make_type_class sub,

                  superclass = make_type_class super};

fun make_classrel_clauses thy subs supers =

  map make_classrel_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 =

  TyVar of name |

  TyFree of name |

  TyConstr of name * combtyp list

datatype combterm =

  CombConst of name * combtyp * combtyp list (* Const and Free *) |

  CombVar of name * combtyp |

  CombApp of combterm * combterm

datatype literal = Literal of bool * combterm

datatype hol_clause =

  HOLClause of {clause_id: int, axiom_name: string, th: thm, kind: kind,

                literals: 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 (TyConstr (_, [_, tp2])) = tp2

  | result_type _ = raise Fail "non-function type"

fun type_of_combterm (CombConst (_, tp, _)) = tp

  | type_of_combterm (CombVar (_, tp)) = tp

  | type_of_combterm (CombApp (t1, _)) = result_type (type_of_combterm 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

      (TyConstr (`make_fixed_type_const a, folTypes), ts)

    end

  | type_of (tp as TFree (a, _)) = (TyFree (`make_fixed_type_var a), [tp])

  | type_of (tp as TVar (x, _)) =

    (TyVar (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)) =

      TyConstr (`make_fixed_type_const a, map simp_type_of Ts)

  | simp_type_of (TFree (a, _)) = TyFree (`make_fixed_type_var a)

  | simp_type_of (TVar (x, _)) =

    TyVar (make_schematic_type_var x, string_of_indexname x)

(* convert a Term.term (with combinators) into a combterm, also accummulate sort info *)

fun combterm_of 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_of _ (Free(v, T)) =

      let val (tp,ts) = type_of T

          val v' = CombConst (`make_fixed_var v, tp, [])

      in  (v',ts)  end

  | combterm_of _ (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_of thy (P $ Q) =

      let val (P', tsP) = combterm_of thy P

          val (Q', tsQ) = combterm_of thy Q

      in  (CombApp (P', Q'), union (op =) tsP tsQ)  end

  | combterm_of _ (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_of 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

      (Literal (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;