theory Fv+
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imports "Nominal2_Atoms" "Abs" "Perm" "Rsp"+
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begin+
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+
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(* Bindings are a list of lists of lists of triples.+
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+
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   The first list represents the datatypes defined.+
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   The second list represents the constructors.+
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   The internal list is a list of all the bndings that+
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   concern the constructor.+
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+
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   Every triple consists of a function, the binding and+
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   the body.+
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+
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  Eg:+
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nominal_datatype+
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+
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   C1+
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 | C2 x y z bind x in z+
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 | C3 x y z bind f x in z bind g y in z+
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+
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yields:+
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[+
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 [],+
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 [(NONE, 0, 2)],+
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 [(SOME (Const f), 0, 2), (Some (Const g), 1, 2)]]+
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+
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A SOME binding has to have a function which takes an appropriate+
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argument and returns an atom set. A NONE binding has to be on an+
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argument that is an atom or an atom set.+
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+
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How the procedure works:+
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  For each of the defined datatypes,+
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  For each of the constructors,+
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  It creates a union of free variables for each argument.+
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+
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  For an argument the free variables are the variables minus+
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  bound variables.+
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+
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  The variables are:+
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    For an atom, a singleton set with the atom itself.+
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    For an atom set, the atom set itself.+
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    For a recursive argument, the appropriate fv function applied to it.+
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    (* TODO: This one is not implemented *)+
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    For other arguments it should be an appropriate fv function stored+
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      in the database.+
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  The bound variables are a union of results of all bindings that+
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  involve the given argument. For a paricular binding the result is:+
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    For a function applied to an argument this function with the argument.+
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    For an atom, a singleton set with the atom itself.+
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    For an atom set, the atom set itself.+
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    For a recursive argument, the appropriate fv function applied to it.+
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    (* TODO: This one is not implemented *)+
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    For other arguments it should be an appropriate fv function stored+
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      in the database.+
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+
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  For every argument that is a binding, we add a the same binding in its+
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  body.+
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*)+
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+
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ML {*+
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fun is_atom thy typ =+
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  Sign.of_sort thy (typ, @{sort at})+
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+
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fun is_atom_set thy (Type ("fun", [t, @{typ bool}])) = is_atom thy t+
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  | is_atom_set thy _ = false;+
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*}+
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+
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+
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+
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(* Like map2, only if the second list is empty passes empty lists insted of error *)+
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ML {*+
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fun map2i _ [] [] = []+
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  | map2i f (x :: xs) (y :: ys) = f x y :: map2i f xs ys+
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  | map2i f (x :: xs) [] = f x [] :: map2i f xs []+
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  | map2i _ _ _ = raise UnequalLengths;+
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*}+
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+
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(* Finds bindings with the same function and binding, and gathers all+
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   bodys for such pairs+
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 *)+
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ML {*+
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fun gather_binds binds =+
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let+
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  fun gather_binds_cons binds =+
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    let+
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      val common = map (fn (f, bi, _) => (f, bi)) binds+
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      val nodups = distinct (op =) common+
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      fun find_bodys (sf, sbi) =+
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        filter (fn (f, bi, _) => f = sf andalso bi = sbi) binds+
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      val bodys = map ((map (fn (_, _, bo) => bo)) o find_bodys) nodups+
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    in+
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      nodups ~~ bodys+
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    end+
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in+
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  map (map gather_binds_cons) binds+
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end+
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*}+
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+
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ML {*+
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fun un_gather_binds_cons binds =+
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  flat (map (fn (((f, bi), bos), pi) => map (fn bo => ((f, bi, bo), pi)) bos) binds)+
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*}+
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+
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ML {*+
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  open Datatype_Aux; (* typ_of_dtyp, DtRec, ... *);+
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  (* TODO: It is the same as one in 'nominal_atoms' *)+
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  fun mk_atom ty = Const (@{const_name atom}, ty --> @{typ atom});+
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  val noatoms = @{term "{} :: atom set"};+
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  fun mk_single_atom x = HOLogic.mk_set @{typ atom} [mk_atom (type_of x) $ x];+
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  fun mk_union sets =+
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    fold (fn a => fn b =>+
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      if a = noatoms then b else+
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      if b = noatoms then a else+
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      if a = b then a else+
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      HOLogic.mk_binop @{const_name sup} (a, b)) (rev sets) noatoms;+
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  val mk_inter = foldr1 (HOLogic.mk_binop @{const_name inf})+
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  fun mk_conjl props =+
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    fold (fn a => fn b =>+
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      if a = @{term True} then b else+
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      if b = @{term True} then a else+
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      HOLogic.mk_conj (a, b)) props @{term True};+
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  fun mk_diff a b =+
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    if b = noatoms then a else+
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    if b = a then noatoms else+
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    HOLogic.mk_binop @{const_name minus} (a, b);+
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  fun mk_atoms t =+
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    let+
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      val ty = fastype_of t;+
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      val atom_ty = HOLogic.dest_setT ty --> @{typ atom};+
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      val img_ty = atom_ty --> ty --> @{typ "atom set"};+
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    in+
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      (Const (@{const_name image}, img_ty) $ Const (@{const_name atom}, atom_ty) $ t)+
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    end;+
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  (* Similar to one in USyntax *)+
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  fun mk_pair (fst, snd) =+
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    let val ty1 = fastype_of fst+
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      val ty2 = fastype_of snd+
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      val c = HOLogic.pair_const ty1 ty2+
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    in c $ fst $ snd+
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    end;+
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+
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*}+
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+
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ML {* fun add_perm (p1, p2) = Const(@{const_name plus}, @{typ "perm \<Rightarrow> perm \<Rightarrow> perm"}) $ p1 $ p2 *}+
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+
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ML {*+
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fun non_rec_binds l =+
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let+
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  fun is_non_rec (SOME (f, false), _, _) = SOME f+
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    | is_non_rec _ = NONE+
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in+
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  distinct (op =) (map_filter is_non_rec (flat (flat l)))+
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end+
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*}+
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+
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(* We assume no bindings in the type on which bn is defined *)+
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ML {*+
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fun fv_bn thy (dt_info : Datatype_Aux.info) fv_frees (bn, ith_dtyp, args_in_bn) =+
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let+
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  val {descr, sorts, ...} = dt_info;+
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  fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);+
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  val fvbn_name = "fv_" ^ (Long_Name.base_name (fst (dest_Const bn)));+
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  val fvbn = Free (fvbn_name, fastype_of (nth fv_frees ith_dtyp));+
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  fun fv_bn_constr (cname, dts) =+
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  let+
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    val Ts = map (typ_of_dtyp descr sorts) dts;+
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    val names = Datatype_Prop.make_tnames Ts;+
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    val args = map Free (names ~~ Ts);+
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    val c = Const (cname, Ts ---> (nth_dtyp ith_dtyp));+
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    fun fv_arg ((dt, x), arg_no) =+
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      let+
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        val ty = fastype_of x+
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      in+
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        if arg_no mem args_in_bn then +
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          (if is_rec_type dt then+
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            (if body_index dt = ith_dtyp then fvbn $ x else error "fv_bn: not good")+
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          else @{term "{} :: atom set"}) else+
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        if is_atom thy ty then mk_single_atom x else+
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        if is_atom_set thy ty then mk_atoms x else+
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        if is_rec_type dt then nth fv_frees (body_index dt) $ x else+
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        @{term "{} :: atom set"}+
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      end;+
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    val arg_nos = 0 upto (length dts - 1)+
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  in+
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    HOLogic.mk_Trueprop (HOLogic.mk_eq+
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      (fvbn $ list_comb (c, args), mk_union (map fv_arg (dts ~~ args ~~ arg_nos))))+
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  end;+
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  val (_, (_, _, constrs)) = nth descr ith_dtyp;+
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  val eqs = map fv_bn_constr constrs+
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in+
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  ((bn, fvbn), (fvbn_name, eqs))+
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end+
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*}+
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+
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ML {* fn x => split_list(flat x) *}+
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ML {* fn x => apsnd flat (split_list (map split_list x)) *}+
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(* TODO: Notice datatypes without bindings and replace alpha with equality *)+
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ML {*+
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fun define_fv_alpha (dt_info : Datatype_Aux.info) bindsall bns lthy =+
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let+
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  val thy = ProofContext.theory_of lthy;+
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  val {descr, sorts, ...} = dt_info;+
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  fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);+
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  val fv_names = Datatype_Prop.indexify_names (map (fn (i, _) =>+
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    "fv_" ^ name_of_typ (nth_dtyp i)) descr);+
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  val fv_types = map (fn (i, _) => nth_dtyp i --> @{typ "atom set"}) descr;+
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  val fv_frees = map Free (fv_names ~~ fv_types);+
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  val nr_bns = non_rec_binds bindsall;+
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  val rel_bns = filter (fn (bn, _, _) => bn mem nr_bns) bns;+
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  val (bn_fv_bns, fv_bn_names_eqs) = split_list (map (fv_bn thy dt_info fv_frees) rel_bns);+
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  val fv_bns = map snd bn_fv_bns;+
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  val (fv_bn_names, fv_bn_eqs) = split_list fv_bn_names_eqs;+
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  val alpha_names = Datatype_Prop.indexify_names (map (fn (i, _) =>+
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    "alpha_" ^ name_of_typ (nth_dtyp i)) descr);+
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  val alpha_types = map (fn (i, _) => nth_dtyp i --> nth_dtyp i --> @{typ bool}) descr;+
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  val alpha_frees = map Free (alpha_names ~~ alpha_types);+
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  fun fv_alpha_constr ith_dtyp (cname, dts) bindcs =+
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    let+
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      val Ts = map (typ_of_dtyp descr sorts) dts;+
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      val bindslen = length bindcs+
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      val pi_strs_same = replicate bindslen "pi"+
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      val pi_strs = Name.variant_list [] pi_strs_same;+
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      val pis = map (fn ps => Free (ps, @{typ perm})) pi_strs;+
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      val bind_pis_gath = bindcs ~~ pis;+
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      val bind_pis = un_gather_binds_cons bind_pis_gath;+
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      val bindcs = map fst bind_pis;+
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      val names = Name.variant_list pi_strs (Datatype_Prop.make_tnames Ts);+
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      val args = map Free (names ~~ Ts);+
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      val names2 = Name.variant_list (pi_strs @ names) (Datatype_Prop.make_tnames Ts);+
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      val args2 = map Free (names2 ~~ Ts);+
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      val c = Const (cname, Ts ---> (nth_dtyp ith_dtyp));+
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      val fv_c = nth fv_frees ith_dtyp;+
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      val alpha = nth alpha_frees ith_dtyp;+
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      val arg_nos = 0 upto (length dts - 1)+
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      fun fv_bind args (NONE, i, _) =+
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            if is_rec_type (nth dts i) then (nth fv_frees (body_index (nth dts i))) $ (nth args i) else+
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            if ((is_atom thy) o fastype_of) (nth args i) then mk_single_atom (nth args i) else+
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            if ((is_atom_set thy) o fastype_of) (nth args i) then mk_atoms (nth args i) else+
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            (* TODO we do not know what to do with non-atomizable things *)+
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            @{term "{} :: atom set"}+
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        | fv_bind args (SOME (f, _), i, _) = f $ (nth args i);+
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      fun fv_binds args relevant = mk_union (map (fv_bind args) relevant)+
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      fun find_nonrec_binder j (SOME (f, false), i, _) = if i = j then SOME f else NONE+
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        | find_nonrec_binder _ _ = NONE+
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      fun fv_arg ((dt, x), arg_no) =+
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        case get_first (find_nonrec_binder arg_no) bindcs of+
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          SOME f =>+
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            (case get_first (fn (x, y) => if x = f then SOME y else NONE) bn_fv_bns of+
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                SOME fv_bn => fv_bn $ x+
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              | NONE => error "bn specified in a non-rec binding but not in bn list")+
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        | NONE =>+
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            let+
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              val arg =+
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                if is_rec_type dt then nth fv_frees (body_index dt) $ x else+
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                if ((is_atom thy) o fastype_of) x then mk_single_atom x else+
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                if ((is_atom_set thy) o fastype_of) x then mk_atoms x else+
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                (* TODO we do not know what to do with non-atomizable things *)+
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                @{term "{} :: atom set"}+
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              (* If i = j then we generate it only once *)+
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              val relevant = filter (fn (_, i, j) => ((i = arg_no) orelse (j = arg_no))) bindcs;+
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              val sub = fv_binds args relevant+
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            in+
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              mk_diff arg sub+
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            end;+
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      val fv_eq = HOLogic.mk_Trueprop (HOLogic.mk_eq+
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        (fv_c $ list_comb (c, args), mk_union (map fv_arg  (dts ~~ args ~~ arg_nos))))+
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      val alpha_rhs =+
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        HOLogic.mk_Trueprop (alpha $ (list_comb (c, args)) $ (list_comb (c, args2)));+
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      fun alpha_arg ((dt, arg_no), (arg, arg2)) =+
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        let+
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          val relevant = filter (fn ((_, i, j), _) => i = arg_no orelse j = arg_no) bind_pis;+
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        in+
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          if relevant = [] then (+
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            if is_rec_type dt then (nth alpha_frees (body_index dt) $ arg $ arg2)+
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            else (HOLogic.mk_eq (arg, arg2)))+
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          else+
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            if is_rec_type dt then let+
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              (* THE HARD CASE *)+
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              val (rbinds, rpis) = split_list relevant+
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              val lhs_binds = fv_binds args rbinds+
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              val lhs = mk_pair (lhs_binds, arg);+
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              val rhs_binds = fv_binds args2 rbinds;+
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              val rhs = mk_pair (rhs_binds, arg2);+
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              val alpha = nth alpha_frees (body_index dt);+
−
              val fv = nth fv_frees (body_index dt);+
−
              val pi = foldr1 add_perm (distinct (op =) rpis);+
−
              val alpha_gen_pre = Const (@{const_name alpha_gen}, dummyT) $ lhs $ alpha $ fv $ pi $ rhs;+
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              val alpha_gen = Syntax.check_term lthy alpha_gen_pre+
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(*              val pi_supps = map ((curry op $) @{term "supp :: perm \<Rightarrow> atom set"}) rpis;+
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              val pi_supps_eq = HOLogic.mk_eq (mk_inter pi_supps, @{term "{} :: atom set"}) *)+
−
            in+
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              (*if length pi_supps > 1 then HOLogic.mk_conj (alpha_gen, pi_supps_eq) else*)+
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              alpha_gen+
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            (* TODO Add some test that is makes sense *)+
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            end else @{term "True"}+
−
        end+
−
      val alphas = map alpha_arg (dts ~~ arg_nos ~~ (args ~~ args2))+
−
      val alpha_lhss = mk_conjl alphas+
−
      val alpha_lhss_ex =+
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        fold (fn pi_str => fn t => HOLogic.mk_exists (pi_str, @{typ perm}, t)) pi_strs alpha_lhss+
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      val alpha_eq = Logic.mk_implies (HOLogic.mk_Trueprop alpha_lhss_ex, alpha_rhs)+
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    in+
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      (fv_eq, alpha_eq)+
−
    end;+
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  fun fv_alpha_eq (i, (_, _, constrs)) binds = map2i (fv_alpha_constr i) constrs binds;+
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  val fveqs_alphaeqs = map2i fv_alpha_eq descr (gather_binds bindsall)+
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  val (fv_eqs_perfv, alpha_eqs) = apsnd flat (split_list (map split_list fveqs_alphaeqs))+
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  val rel_bns_nos = map (fn (_, i, _) => i) rel_bns;+
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  fun filter_fun (a, b) = b mem rel_bns_nos;+
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  val all_fvs = (fv_names ~~ fv_eqs_perfv) ~~ (0 upto (length fv_names - 1))+
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  val (fv_names_fst, fv_eqs_fst) = apsnd flat (split_list (map fst (filter_out filter_fun all_fvs)))+
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  val (fv_names_snd, fv_eqs_snd) = apsnd flat (split_list (map fst (filter filter_fun all_fvs)))+
−
  val fv_eqs_all = fv_eqs_fst @ (flat fv_bn_eqs);+
−
  val fv_names_all = fv_names_fst @ fv_bn_names;+
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  val add_binds = map (fn x => (Attrib.empty_binding, x))+
−
(* Function_Fun.add_fun Function_Common.default_config ... true *)+
−
  val _ = map tracing (map (Syntax.string_of_term @{context}) fv_eqs_all)+
−
  val (fvs, lthy') = (Primrec.add_primrec+
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    (map (fn s => (Binding.name s, NONE, NoSyn)) fv_names_all) (add_binds fv_eqs_all) lthy)+
−
  val (fvs2, lthy'') =+
−
    if fv_eqs_snd = [] then (([], []), lthy') else+
−
   (Primrec.add_primrec+
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    (map (fn s => (Binding.name s, NONE, NoSyn)) fv_names_snd) (add_binds fv_eqs_snd) lthy')+
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  val (alphas, lthy''') = (Inductive.add_inductive_i+
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     {quiet_mode = true, verbose = false, alt_name = Binding.empty,+
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      coind = false, no_elim = false, no_ind = false, skip_mono = true, fork_mono = false}+
−
     (map2 (fn x => fn y => ((Binding.name x, y), NoSyn)) alpha_names alpha_types) []+
−
     (add_binds alpha_eqs) [] lthy'')+
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in+
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  ((fvs, alphas), lthy''')+
−
end+
−
*}+
−
+
−
(*+
−
atom_decl name+
−
+
−
datatype lam =+
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  VAR "name"+
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| APP "lam" "lam"+
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| LET "bp" "lam"+
−
and bp =+
−
  BP "name" "lam"+
−
+
−
primrec+
−
  bi::"bp \<Rightarrow> atom set"+
−
where+
−
  "bi (BP x t) = {atom x}"+
−
+
−
setup {* snd o define_raw_perms (Datatype.the_info @{theory} "Fv.lam") 2 *}+
−
+
−
+
−
local_setup {*+
−
  snd o define_fv_alpha (Datatype.the_info @{theory} "Fv.lam")+
−
  [[[], [], [(SOME (@{term bi}, false), 0, 1)]], [[]]] [(@{term bi}, 1, [0])] *}+
−
print_theorems+
−
*)+
−
+
−
(*+
−
datatype rtrm1 =+
−
  rVr1 "name"+
−
| rAp1 "rtrm1" "rtrm1"+
−
| rLm1 "name" "rtrm1"        --"name is bound in trm1"+
−
| rLt1 "bp" "rtrm1" "rtrm1"   --"all variables in bp are bound in the 2nd trm1"+
−
and bp =+
−
  BUnit+
−
| BVr "name"+
−
| BPr "bp" "bp"+
−
+
−
(* to be given by the user *)+
−
+
−
primrec+
−
  bv1+
−
where+
−
  "bv1 (BUnit) = {}"+
−
| "bv1 (BVr x) = {atom x}"+
−
| "bv1 (BPr bp1 bp2) = (bv1 bp1) \<union> (bv1 bp1)"+
−
+
−
setup {* snd o define_raw_perms ["rtrm1", "bp"] ["Fv.rtrm1", "Fv.bp"] *}+
−
+
−
local_setup {* define_fv_alpha "Fv.rtrm1"+
−
  [[[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]], [[(SOME @{term bv1}, 0)], [], [(SOME @{term bv1}, 0)]]],+
−
   [[], [[]], [[], []]]] *}+
−
print_theorems+
−
*)+
−
+
−
+
−
ML {*+
−
fun alpha_inj_tac dist_inj intrs elims =+
−
  SOLVED' (asm_full_simp_tac (HOL_ss addsimps intrs)) ORELSE'+
−
  (rtac @{thm iffI} THEN' RANGE [+
−
     (eresolve_tac elims THEN_ALL_NEW+
−
       asm_full_simp_tac (HOL_ss addsimps dist_inj)+
−
     ),+
−
     asm_full_simp_tac (HOL_ss addsimps intrs)])+
−
*}+
−
+
−
ML {*+
−
fun build_alpha_inj_gl thm =+
−
  let+
−
    val prop = prop_of thm;+
−
    val concl = HOLogic.dest_Trueprop (Logic.strip_imp_concl prop);+
−
    val hyps = map HOLogic.dest_Trueprop (Logic.strip_imp_prems prop);+
−
    fun list_conj l = foldr1 HOLogic.mk_conj l;+
−
  in+
−
    if hyps = [] then concl+
−
    else HOLogic.mk_eq (concl, list_conj hyps)+
−
  end;+
−
*}+
−
+
−
ML {*+
−
fun build_alpha_inj intrs dist_inj elims ctxt =+
−
let+
−
  val ((_, thms_imp), ctxt') = Variable.import false intrs ctxt;+
−
  val gls = map (HOLogic.mk_Trueprop o build_alpha_inj_gl) thms_imp;+
−
  fun tac _ = alpha_inj_tac dist_inj intrs elims 1;+
−
  val thms = map (fn gl => Goal.prove ctxt' [] [] gl tac) gls;+
−
in+
−
  Variable.export ctxt' ctxt thms+
−
end+
−
*}+
−
+
−
ML {*+
−
fun build_alpha_refl_gl alphas (x, y, z) =+
−
let+
−
  fun build_alpha alpha =+
−
    let+
−
      val ty = domain_type (fastype_of alpha);+
−
      val var = Free(x, ty);+
−
      val var2 = Free(y, ty);+
−
      val var3 = Free(z, ty);+
−
      val symp = HOLogic.mk_imp (alpha $ var $ var2, alpha $ var2 $ var);+
−
      val transp = HOLogic.mk_imp (alpha $ var $ var2,+
−
        HOLogic.mk_all (z, ty,+
−
          HOLogic.mk_imp (alpha $ var2 $ var3, alpha $ var $ var3)))+
−
    in+
−
      ((alpha $ var $ var), (symp, transp))+
−
    end;+
−
  val (refl_eqs, eqs) = split_list (map build_alpha alphas)+
−
  val (sym_eqs, trans_eqs) = split_list eqs+
−
  fun conj l = @{term Trueprop} $ foldr1 HOLogic.mk_conj l+
−
in+
−
  (conj refl_eqs, (conj sym_eqs, conj trans_eqs))+
−
end+
−
*}+
−
+
−
ML {*+
−
fun reflp_tac induct inj ctxt =+
−
  rtac induct THEN_ALL_NEW+
−
  simp_tac ((mk_minimal_ss ctxt) addsimps inj) THEN_ALL_NEW+
−
  split_conjs THEN_ALL_NEW REPEAT o rtac @{thm exI[of _ "0 :: perm"]}+
−
  THEN_ALL_NEW split_conjs THEN_ALL_NEW asm_full_simp_tac (HOL_ss addsimps+
−
     @{thms alpha_gen fresh_star_def fresh_zero_perm permute_zero ball_triv+
−
       add_0_left supp_zero_perm Int_empty_left})+
−
*}+
−
+
−
+
−
lemma exi_neg: "\<exists>(pi :: perm). P pi \<Longrightarrow> (\<And>(p :: perm). P p \<Longrightarrow> Q (- p)) \<Longrightarrow> \<exists>pi. Q pi"+
−
apply (erule exE)+
−
apply (rule_tac x="-pi" in exI)+
−
by auto+
−
+
−
ML {*+
−
fun symp_tac induct inj eqvt ctxt =+
−
  ind_tac induct THEN_ALL_NEW+
−
  simp_tac ((mk_minimal_ss ctxt) addsimps inj) THEN_ALL_NEW split_conjs+
−
  THEN_ALL_NEW+
−
  REPEAT o etac @{thm exi_neg}+
−
  THEN_ALL_NEW+
−
  split_conjs THEN_ALL_NEW+
−
  asm_full_simp_tac (HOL_ss addsimps @{thms supp_minus_perm minus_add[symmetric]}) THEN_ALL_NEW+
−
  (rtac @{thm alpha_gen_compose_sym} THEN' RANGE [+
−
    (asm_full_simp_tac (HOL_ss addsimps @{thms plus_perm_eq})),+
−
    (asm_full_simp_tac (HOL_ss addsimps (eqvt @ all_eqvts ctxt)))+
−
  ])+
−
*}+
−
+
−
ML {*+
−
fun imp_elim_tac case_rules =+
−
  Subgoal.FOCUS (fn {concl, context, ...} =>+
−
    case term_of concl of+
−
      _ $ (_ $ asm $ _) =>+
−
        let+
−
          fun filter_fn case_rule = (+
−
            case Logic.strip_assums_hyp (prop_of case_rule) of+
−
              ((_ $ asmc) :: _) =>+
−
                let+
−
                  val thy = ProofContext.theory_of context+
−
                in+
−
                  Pattern.matches thy (asmc, asm)+
−
                end+
−
            | _ => false)+
−
          val matching_rules = filter filter_fn case_rules+
−
        in+
−
         (rtac impI THEN' rotate_tac (~1) THEN' eresolve_tac matching_rules) 1+
−
        end+
−
    | _ => no_tac+
−
  )+
−
*}+
−
+
−
+
−
lemma exi_sum: "\<exists>(pi :: perm). P pi \<Longrightarrow> \<exists>(pi :: perm). Q pi \<Longrightarrow> (\<And>(p :: perm) (pi :: perm). P p \<Longrightarrow> Q pi \<Longrightarrow> R (pi + p)) \<Longrightarrow> \<exists>pi. R pi"+
−
apply (erule exE)++
−
apply (rule_tac x="pia + pi" in exI)+
−
by auto+
−
+
−
ML {*+
−
fun is_ex (Const ("Ex", _) $ Abs _) = true+
−
  | is_ex _ = false;+
−
*}+
−
+
−
ML {*+
−
fun eetac rule = Subgoal.FOCUS_PARAMS +
−
  (fn (focus) =>+
−
     let+
−
       val concl = #concl focus+
−
       val prems = Logic.strip_imp_prems (term_of concl)+
−
       val exs = filter (fn x => is_ex (HOLogic.dest_Trueprop x)) prems+
−
       val cexs = map (SOME o (cterm_of (ProofContext.theory_of (#context focus)))) exs+
−
       val thins = map (fn cex => Drule.instantiate' [] [cex] Drule.thin_rl) cexs+
−
     in+
−
     (etac rule THEN' RANGE[+
−
        atac,+
−
        eresolve_tac thins+
−
     ]) 1+
−
     end+
−
  )+
−
*}+
−
+
−
ML {*+
−
fun transp_tac ctxt induct alpha_inj term_inj distinct cases eqvt =+
−
  ind_tac induct THEN_ALL_NEW+
−
  (TRY o rtac allI THEN' imp_elim_tac cases ctxt) THEN_ALL_NEW+
−
  asm_full_simp_tac ((mk_minimal_ss ctxt) addsimps alpha_inj) THEN_ALL_NEW+
−
  split_conjs THEN_ALL_NEW REPEAT o (eetac @{thm exi_sum} ctxt) THEN_ALL_NEW split_conjs+
−
  THEN_ALL_NEW (asm_full_simp_tac (HOL_ss addsimps (term_inj @ distinct)))+
−
  THEN_ALL_NEW split_conjs THEN_ALL_NEW+
−
  TRY o (etac @{thm alpha_gen_compose_trans} THEN' RANGE[atac]) THEN_ALL_NEW+
−
  (asm_full_simp_tac (HOL_ss addsimps (all_eqvts ctxt @ eqvt @ term_inj @ distinct)))+
−
*}+
−
+
−
lemma transp_aux:+
−
  "(\<And>xa ya. R xa ya \<longrightarrow> (\<forall>z. R ya z \<longrightarrow> R xa z)) \<Longrightarrow> transp R"+
−
  unfolding transp_def+
−
  by blast+
−
+
−
ML {*+
−
fun equivp_tac reflps symps transps =+
−
  simp_tac (HOL_ss addsimps @{thms equivp_reflp_symp_transp reflp_def symp_def})+
−
  THEN' rtac conjI THEN' rtac allI THEN'+
−
  resolve_tac reflps THEN'+
−
  rtac conjI THEN' rtac allI THEN' rtac allI THEN'+
−
  resolve_tac symps THEN'+
−
  rtac @{thm transp_aux} THEN' resolve_tac transps+
−
*}+
−
+
−
ML {*+
−
fun build_equivps alphas term_induct alpha_induct term_inj alpha_inj distinct cases eqvt ctxt =+
−
let+
−
  val ([x, y, z], ctxt') = Variable.variant_fixes ["x","y","z"] ctxt;+
−
  val (reflg, (symg, transg)) = build_alpha_refl_gl alphas (x, y, z)+
−
  fun reflp_tac' _ = reflp_tac term_induct alpha_inj ctxt 1;+
−
  fun symp_tac' _ = symp_tac alpha_induct alpha_inj eqvt ctxt 1;+
−
  fun transp_tac' _ = transp_tac ctxt alpha_induct alpha_inj term_inj distinct cases eqvt 1;+
−
  val reflt = Goal.prove ctxt' [] [] reflg reflp_tac';+
−
  val symt = Goal.prove ctxt' [] [] symg symp_tac';+
−
  val transt = Goal.prove ctxt' [] [] transg transp_tac';+
−
  val [refltg, symtg, transtg] = Variable.export ctxt' ctxt [reflt, symt, transt]+
−
  val reflts = HOLogic.conj_elims refltg+
−
  val symts = HOLogic.conj_elims symtg+
−
  val transts = HOLogic.conj_elims transtg+
−
  fun equivp alpha =+
−
    let+
−
      val equivp = Const (@{const_name equivp}, fastype_of alpha --> @{typ bool})+
−
      val goal = @{term Trueprop} $ (equivp $ alpha)+
−
      fun tac _ = equivp_tac reflts symts transts 1+
−
    in+
−
      Goal.prove ctxt [] [] goal tac+
−
    end+
−
in+
−
  map equivp alphas+
−
end+
−
*}+
−
+
−
(*+
−
Tests:+
−
prove alpha1_reflp_aux: {* fst (build_alpha_refl_gl [@{term alpha_rtrm1}, @{term alpha_bp}] ("x","y","z")) *}+
−
by (tactic {* reflp_tac @{thm rtrm1_bp.induct} @{thms alpha1_inj} 1 *})+
−
+
−
prove alpha1_symp_aux: {* (fst o snd) (build_alpha_refl_gl [@{term alpha_rtrm1}, @{term alpha_bp}] ("x","y","z")) *}+
−
by (tactic {* symp_tac @{thm alpha_rtrm1_alpha_bp.induct} @{thms alpha1_inj} @{thms alpha1_eqvt} 1 *})+
−
+
−
prove alpha1_transp_aux: {* (snd o snd) (build_alpha_refl_gl [@{term alpha_rtrm1}, @{term alpha_bp}] ("x","y","z")) *}+
−
by (tactic {* transp_tac @{context} @{thm alpha_rtrm1_alpha_bp.induct} @{thms alpha1_inj} @{thms rtrm1.inject bp.inject} @{thms rtrm1.distinct bp.distinct} @{thms alpha_rtrm1.cases alpha_bp.cases} @{thms alpha1_eqvt} 1 *})+
−
+
−
lemma alpha1_equivp:+
−
  "equivp alpha_rtrm1"+
−
  "equivp alpha_bp"+
−
apply (tactic {*+
−
  (simp_tac (HOL_ss addsimps @{thms equivp_reflp_symp_transp reflp_def symp_def})+
−
  THEN' rtac @{thm conjI} THEN' rtac @{thm allI} THEN'+
−
  resolve_tac (HOLogic.conj_elims @{thm alpha1_reflp_aux})+
−
  THEN' rtac @{thm conjI} THEN' rtac @{thm allI} THEN' rtac @{thm allI} THEN'+
−
  resolve_tac (HOLogic.conj_elims @{thm alpha1_symp_aux}) THEN' rtac @{thm transp_aux}+
−
  THEN' resolve_tac (HOLogic.conj_elims @{thm alpha1_transp_aux})+
−
)+
−
1 *})+
−
done*)+
−
+
−
ML {*+
−
fun dtyp_no_of_typ _ (TFree (n, _)) = error "dtyp_no_of_typ: Illegal free"+
−
  | dtyp_no_of_typ _ (TVar _) = error "dtyp_no_of_typ: Illegal schematic"+
−
  | dtyp_no_of_typ dts (Type (tname, Ts)) =+
−
      case try (find_index (curry op = tname o fst)) dts of+
−
        NONE => error "dtyp_no_of_typ: Illegal recursion"+
−
      | SOME i => i+
−
*}+
−
+
−
end+
−