TFL/usyntax.ML

add lemma complete_algebra_summable_geometric

+
−(*  Title:      TFL/usyntax.ML
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−    ID:         $Id$
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−    Author:     Konrad Slind, Cambridge University Computer Laboratory
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−    Copyright   1997  University of Cambridge
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−
+
−Emulation of HOL's abstract syntax functions.
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−*)
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−
+
−signature USYNTAX =
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−sig
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−  datatype lambda = VAR   of {Name : string, Ty : typ}
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−                  | CONST of {Name : string, Ty : typ}
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−                  | COMB  of {Rator: term, Rand : term}
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−                  | LAMB  of {Bvar : term, Body : term}
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−
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−  val alpha : typ
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−
+
−  (* Types *)
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−  val type_vars  : typ -> typ list
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−  val type_varsl : typ list -> typ list
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−  val mk_vartype : string -> typ
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−  val is_vartype : typ -> bool
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−  val strip_prod_type : typ -> typ list
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−
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−  (* Terms *)
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−  val free_vars_lr : term -> term list
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−  val type_vars_in_term : term -> typ list
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−  val dest_term  : term -> lambda
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−
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−  (* Prelogic *)
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−  val inst      : (typ*typ) list -> term -> term
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−
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−  (* Construction routines *)
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−  val mk_abs    :{Bvar  : term, Body : term} -> term
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−
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−  val mk_imp    :{ant : term, conseq :  term} -> term
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−  val mk_select :{Bvar : term, Body : term} -> term
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−  val mk_forall :{Bvar : term, Body : term} -> term
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−  val mk_exists :{Bvar : term, Body : term} -> term
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−  val mk_conj   :{conj1 : term, conj2 : term} -> term
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−  val mk_disj   :{disj1 : term, disj2 : term} -> term
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−  val mk_pabs   :{varstruct : term, body : term} -> term
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−
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−  (* Destruction routines *)
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−  val dest_const: term -> {Name : string, Ty : typ}
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−  val dest_comb : term -> {Rator : term, Rand : term}
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−  val dest_abs  : string list -> term -> {Bvar : term, Body : term} * string list
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−  val dest_eq     : term -> {lhs : term, rhs : term}
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−  val dest_imp    : term -> {ant : term, conseq : term}
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−  val dest_forall : term -> {Bvar : term, Body : term}
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−  val dest_exists : term -> {Bvar : term, Body : term}
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−  val dest_neg    : term -> term
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−  val dest_conj   : term -> {conj1 : term, conj2 : term}
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−  val dest_disj   : term -> {disj1 : term, disj2 : term}
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−  val dest_pair   : term -> {fst : term, snd : term}
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−  val dest_pabs   : string list -> term -> {varstruct : term, body : term, used : string list}
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−
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−  val lhs   : term -> term
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−  val rhs   : term -> term
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−  val rand  : term -> term
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−
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−  (* Query routines *)
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−  val is_imp    : term -> bool
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−  val is_forall : term -> bool
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−  val is_exists : term -> bool
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−  val is_neg    : term -> bool
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−  val is_conj   : term -> bool
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−  val is_disj   : term -> bool
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−  val is_pair   : term -> bool
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−  val is_pabs   : term -> bool
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−
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−  (* Construction of a term from a list of Preterms *)
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−  val list_mk_abs    : (term list * term) -> term
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−  val list_mk_imp    : (term list * term) -> term
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−  val list_mk_forall : (term list * term) -> term
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−  val list_mk_conj   : term list -> term
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−
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−  (* Destructing a term to a list of Preterms *)
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−  val strip_comb     : term -> (term * term list)
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−  val strip_abs      : term -> (term list * term)
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−  val strip_imp      : term -> (term list * term)
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−  val strip_forall   : term -> (term list * term)
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−  val strip_exists   : term -> (term list * term)
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−  val strip_disj     : term -> term list
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−
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−  (* Miscellaneous *)
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−  val mk_vstruct : typ -> term list -> term
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−  val gen_all    : term -> term
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−  val find_term  : (term -> bool) -> term -> term option
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−  val dest_relation : term -> term * term * term
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−  val is_WFR : term -> bool
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−  val ARB : typ -> term
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−end;
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−
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−structure USyntax: USYNTAX =
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−struct
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−
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−infix 4 ##;
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−
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−fun USYN_ERR func mesg = Utils.ERR {module = "USyntax", func = func, mesg = mesg};
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−
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−
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−(*---------------------------------------------------------------------------
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− *
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− *                            Types
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− *
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− *---------------------------------------------------------------------------*)
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−val mk_prim_vartype = TVar;
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−fun mk_vartype s = mk_prim_vartype ((s, 0), HOLogic.typeS);
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−
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−(* But internally, it's useful *)
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−fun dest_vtype (TVar x) = x
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−  | dest_vtype _ = raise USYN_ERR "dest_vtype" "not a flexible type variable";
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−
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−val is_vartype = can dest_vtype;
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−
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−val type_vars  = map mk_prim_vartype o typ_tvars
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−fun type_varsl L = distinct (op =) (fold (curry op @ o type_vars) L []);
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−
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−val alpha  = mk_vartype "'a"
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−val beta   = mk_vartype "'b"
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−
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−val strip_prod_type = HOLogic.prodT_factors;
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−
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−
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−
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−(*---------------------------------------------------------------------------
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− *
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− *                              Terms
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− *
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− *---------------------------------------------------------------------------*)
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−
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−(* Free variables, in order of occurrence, from left to right in the
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− * syntax tree. *)
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−fun free_vars_lr tm =
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−  let fun memb x = let fun m[] = false | m(y::rst) = (x=y)orelse m rst in m end
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−      fun add (t, frees) = case t of
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−            Free   _ => if (memb t frees) then frees else t::frees
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−          | Abs (_,_,body) => add(body,frees)
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−          | f$t =>  add(t, add(f, frees))
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−          | _ => frees
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−  in rev(add(tm,[]))
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−  end;
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−
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−
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−
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−val type_vars_in_term = map mk_prim_vartype o term_tvars;
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−
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−
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−
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−(* Prelogic *)
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−fun dest_tybinding (v,ty) = (#1(dest_vtype v),ty)
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−fun inst theta = subst_vars (map dest_tybinding theta,[])
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−
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−
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−(* Construction routines *)
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−
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−fun mk_abs{Bvar as Var((s,_),ty),Body}  = Abs(s,ty,abstract_over(Bvar,Body))
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−  | mk_abs{Bvar as Free(s,ty),Body}  = Abs(s,ty,abstract_over(Bvar,Body))
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−  | mk_abs _ = raise USYN_ERR "mk_abs" "Bvar is not a variable";
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−
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−
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−fun mk_imp{ant,conseq} =
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−   let val c = Const("op -->",HOLogic.boolT --> HOLogic.boolT --> HOLogic.boolT)
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−   in list_comb(c,[ant,conseq])
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−   end;
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−
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−fun mk_select (r as {Bvar,Body}) =
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−  let val ty = type_of Bvar
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−      val c = Const("Hilbert_Choice.Eps",(ty --> HOLogic.boolT) --> ty)
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−  in list_comb(c,[mk_abs r])
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−  end;
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−
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−fun mk_forall (r as {Bvar,Body}) =
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−  let val ty = type_of Bvar
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−      val c = Const("All",(ty --> HOLogic.boolT) --> HOLogic.boolT)
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−  in list_comb(c,[mk_abs r])
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−  end;
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−
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−fun mk_exists (r as {Bvar,Body}) =
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−  let val ty = type_of Bvar
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−      val c = Const("Ex",(ty --> HOLogic.boolT) --> HOLogic.boolT)
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−  in list_comb(c,[mk_abs r])
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−  end;
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−
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−
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−fun mk_conj{conj1,conj2} =
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−   let val c = Const("op &",HOLogic.boolT --> HOLogic.boolT --> HOLogic.boolT)
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−   in list_comb(c,[conj1,conj2])
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−   end;
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−
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−fun mk_disj{disj1,disj2} =
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−   let val c = Const("op |",HOLogic.boolT --> HOLogic.boolT --> HOLogic.boolT)
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−   in list_comb(c,[disj1,disj2])
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−   end;
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−
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−fun prod_ty ty1 ty2 = HOLogic.mk_prodT (ty1,ty2);
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−
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−local
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−fun mk_uncurry(xt,yt,zt) =
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−    Const("split",(xt --> yt --> zt) --> prod_ty xt yt --> zt)
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−fun dest_pair(Const("Pair",_) $ M $ N) = {fst=M, snd=N}
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−  | dest_pair _ = raise USYN_ERR "dest_pair" "not a pair"
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−fun is_var (Var _) = true | is_var (Free _) = true | is_var _ = false
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−in
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−fun mk_pabs{varstruct,body} =
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− let fun mpa (varstruct, body) =
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−       if is_var varstruct
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−       then mk_abs {Bvar = varstruct, Body = body}
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−       else let val {fst, snd} = dest_pair varstruct
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−            in mk_uncurry (type_of fst, type_of snd, type_of body) $
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−               mpa (fst, mpa (snd, body))
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−            end
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− in mpa (varstruct, body) end
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− handle TYPE _ => raise USYN_ERR "mk_pabs" "";
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−end;
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−
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−(* Destruction routines *)
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−
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−datatype lambda = VAR   of {Name : string, Ty : typ}
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−                | CONST of {Name : string, Ty : typ}
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−                | COMB  of {Rator: term, Rand : term}
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−                | LAMB  of {Bvar : term, Body : term};
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−
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−
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−fun dest_term(Var((s,i),ty)) = VAR{Name = s, Ty = ty}
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−  | dest_term(Free(s,ty))    = VAR{Name = s, Ty = ty}
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−  | dest_term(Const(s,ty))   = CONST{Name = s, Ty = ty}
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−  | dest_term(M$N)           = COMB{Rator=M,Rand=N}
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−  | dest_term(Abs(s,ty,M))   = let  val v = Free(s,ty)
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−                               in LAMB{Bvar = v, Body = Term.betapply (M,v)}
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−                               end
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−  | dest_term(Bound _)       = raise USYN_ERR "dest_term" "Bound";
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−
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−fun dest_const(Const(s,ty)) = {Name = s, Ty = ty}
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−  | dest_const _ = raise USYN_ERR "dest_const" "not a constant";
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−
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−fun dest_comb(t1 $ t2) = {Rator = t1, Rand = t2}
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−  | dest_comb _ =  raise USYN_ERR "dest_comb" "not a comb";
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−
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−fun dest_abs used (a as Abs(s, ty, M)) =
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−     let
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−       val s' = Name.variant used s;
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−       val v = Free(s', ty);
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−     in ({Bvar = v, Body = Term.betapply (a,v)}, s'::used)
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−     end
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−  | dest_abs _ _ =  raise USYN_ERR "dest_abs" "not an abstraction";
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−
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−fun dest_eq(Const("op =",_) $ M $ N) = {lhs=M, rhs=N}
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−  | dest_eq _ = raise USYN_ERR "dest_eq" "not an equality";
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−
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−fun dest_imp(Const("op -->",_) $ M $ N) = {ant=M, conseq=N}
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−  | dest_imp _ = raise USYN_ERR "dest_imp" "not an implication";
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−
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−fun dest_forall(Const("All",_) $ (a as Abs _)) = fst (dest_abs [] a)
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−  | dest_forall _ = raise USYN_ERR "dest_forall" "not a forall";
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−
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−fun dest_exists(Const("Ex",_) $ (a as Abs _)) = fst (dest_abs [] a)
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−  | dest_exists _ = raise USYN_ERR "dest_exists" "not an existential";
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−
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−fun dest_neg(Const("not",_) $ M) = M
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−  | dest_neg _ = raise USYN_ERR "dest_neg" "not a negation";
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−
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−fun dest_conj(Const("op &",_) $ M $ N) = {conj1=M, conj2=N}
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−  | dest_conj _ = raise USYN_ERR "dest_conj" "not a conjunction";
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−
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−fun dest_disj(Const("op |",_) $ M $ N) = {disj1=M, disj2=N}
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−  | dest_disj _ = raise USYN_ERR "dest_disj" "not a disjunction";
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−
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−fun mk_pair{fst,snd} =
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−   let val ty1 = type_of fst
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−       val ty2 = type_of snd
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−       val c = Const("Pair",ty1 --> ty2 --> prod_ty ty1 ty2)
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−   in list_comb(c,[fst,snd])
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−   end;
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−
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−fun dest_pair(Const("Pair",_) $ M $ N) = {fst=M, snd=N}
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−  | dest_pair _ = raise USYN_ERR "dest_pair" "not a pair";
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−
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−
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−local  fun ucheck t = (if #Name(dest_const t) = "split" then t
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−                       else raise Match)
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−in
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−fun dest_pabs used tm =
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−   let val ({Bvar,Body}, used') = dest_abs used tm
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−   in {varstruct = Bvar, body = Body, used = used'}
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−   end handle Utils.ERR _ =>
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−          let val {Rator,Rand} = dest_comb tm
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−              val _ = ucheck Rator
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−              val {varstruct = lv, body, used = used'} = dest_pabs used Rand
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−              val {varstruct = rv, body, used = used''} = dest_pabs used' body
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−          in {varstruct = mk_pair {fst = lv, snd = rv}, body = body, used = used''}
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−          end
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−end;
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−
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−
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−val lhs   = #lhs o dest_eq
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−val rhs   = #rhs o dest_eq
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−val rand  = #Rand o dest_comb
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−
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−
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−(* Query routines *)
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−val is_imp    = can dest_imp
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−val is_forall = can dest_forall
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−val is_exists = can dest_exists
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−val is_neg    = can dest_neg
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−val is_conj   = can dest_conj
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−val is_disj   = can dest_disj
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−val is_pair   = can dest_pair
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−val is_pabs   = can (dest_pabs [])
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−
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−
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−(* Construction of a cterm from a list of Terms *)
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−
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−fun list_mk_abs(L,tm) = fold_rev (fn v => fn M => mk_abs{Bvar=v, Body=M}) L tm;
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−
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−(* These others are almost never used *)
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−fun list_mk_imp(A,c) = fold_rev (fn a => fn tm => mk_imp{ant=a,conseq=tm}) A c;
+
−fun list_mk_forall(V,t) = fold_rev (fn v => fn b => mk_forall{Bvar=v, Body=b})V t;
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−val list_mk_conj = Utils.end_itlist(fn c1 => fn tm => mk_conj{conj1=c1, conj2=tm})
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−
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−
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−(* Need to reverse? *)
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−fun gen_all tm = list_mk_forall(term_frees tm, tm);
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−
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−(* Destructing a cterm to a list of Terms *)
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−fun strip_comb tm =
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−   let fun dest(M$N, A) = dest(M, N::A)
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−         | dest x = x
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−   in dest(tm,[])
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−   end;
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−
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−fun strip_abs(tm as Abs _) =
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−       let val ({Bvar,Body}, _) = dest_abs [] tm
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−           val (bvs, core) = strip_abs Body
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−       in (Bvar::bvs, core)
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−       end
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−  | strip_abs M = ([],M);
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−
+
−
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−fun strip_imp fm =
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−   if (is_imp fm)
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−   then let val {ant,conseq} = dest_imp fm
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−            val (was,wb) = strip_imp conseq
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−        in ((ant::was), wb)
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−        end
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−   else ([],fm);
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−
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−fun strip_forall fm =
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−   if (is_forall fm)
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−   then let val {Bvar,Body} = dest_forall fm
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−            val (bvs,core) = strip_forall Body
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−        in ((Bvar::bvs), core)
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−        end
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−   else ([],fm);
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−
+
−
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−fun strip_exists fm =
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−   if (is_exists fm)
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−   then let val {Bvar, Body} = dest_exists fm
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−            val (bvs,core) = strip_exists Body
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−        in (Bvar::bvs, core)
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−        end
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−   else ([],fm);
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−
+
−fun strip_disj w =
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−   if (is_disj w)
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−   then let val {disj1,disj2} = dest_disj w
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−        in (strip_disj disj1@strip_disj disj2)
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−        end
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−   else [w];
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−
+
−
+
−(* Miscellaneous *)
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−
+
−fun mk_vstruct ty V =
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−  let fun follow_prod_type (Type("*",[ty1,ty2])) vs =
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−              let val (ltm,vs1) = follow_prod_type ty1 vs
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−                  val (rtm,vs2) = follow_prod_type ty2 vs1
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−              in (mk_pair{fst=ltm, snd=rtm}, vs2) end
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−        | follow_prod_type _ (v::vs) = (v,vs)
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−  in #1 (follow_prod_type ty V)  end;
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−
+
−
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−(* Search a term for a sub-term satisfying the predicate p. *)
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−fun find_term p =
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−   let fun find tm =
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−      if (p tm) then SOME tm
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−      else case tm of
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−          Abs(_,_,body) => find body
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−        | (t$u)         => (case find t of NONE => find u | some => some)
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−        | _             => NONE
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−   in find
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−   end;
+
−
+
−fun dest_relation tm =
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−   if (type_of tm = HOLogic.boolT)
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−   then let val (Const("op :",_) $ (Const("Pair",_)$y$x) $ R) = tm
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−        in (R,y,x)
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−        end handle Bind => raise USYN_ERR "dest_relation" "unexpected term structure"
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−   else raise USYN_ERR "dest_relation" "not a boolean term";
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−
+
−fun is_WFR (Const("Wellfounded_Recursion.wf",_)$_) = true
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−  | is_WFR _                 = false;
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−
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−fun ARB ty = mk_select{Bvar=Free("v",ty),
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−                       Body=Const("True",HOLogic.boolT)};
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−
+
−end;