theory Fv+
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imports "../Nominal-General/Nominal2_Atoms" +
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        "Abs" "Perm" "Rsp" "Nominal2_FSet"+
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begin+
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+
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(* The bindings data structure:+
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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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+
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(*+
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An overview of the generation of free variables:+
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+
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1) fv_bn functions are generated only for the non-recursive binds.+
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+
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   An fv_bn for a constructor is a union of values for the arguments:+
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+
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   For an argument x that is in the bn function+
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   - if it is a recursive argument bn' we return: fv_bn' x+
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   - otherwise empty+
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+
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   For an argument x that is not in the bn function+
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   - for atom we return: {atom x}+
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   - for atom set we return: atom ` x+
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   - for a recursive call to type ty' we return: fv_ty' x+
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     with fv of the appropriate type+
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   - otherwise empty+
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+
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2) fv_ty functions generated for all types being defined:+
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+
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   fv_ty for a constructor is a union of values for the arguments.+
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+
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   For an argument that is bound in a shallow binding we return empty.+
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+
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   For an argument x that bound in a non-recursive deep binding+
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   we return: fv_bn x.+
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+
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   Otherwise we return the free variables of the argument minus the+
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   bound variables of the argument.+
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+
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   The free variables for an argument x are:+
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   - for an atom: {atom x}+
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   - for atom set: atom ` x+
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   - for recursive call to type ty' return: fv_ty' x+
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   - for nominal datatype ty' return: fv_ty' x+
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+
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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:+
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+
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   - for a binding function bn: bn x+
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   - for a recursive argument of type ty': fv_fy' x+
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   - for nominal datatype ty' return: fv_ty' x+
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*)+
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+
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(*+
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An overview of the generation of alpha-equivalence:+
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+
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1) alpha_bn relations are generated for binding functions.+
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+
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   An alpha_bn for a constructor is true if a conjunction of+
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   propositions for each argument holds.+
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+
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   For an argument a proposition is build as follows from+
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   th:+
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+
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   - for a recursive argument in the bn function, we return: alpha_bn argl argr+
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   - for a recursive argument for type ty not in bn, we return: alpha_ty argl argr+
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   - for other arguments in the bn function we return: True+
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   - for other arguments not in the bn function we return: argl = argr+
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+
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2) alpha_ty relations are generated for all the types being defined:+
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+
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   For each constructor we gather all the arguments that are bound,+
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   and for each of those we add a permutation. We associate those+
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   permutations with the bindings. Note that two bindings can have+
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   the same permutation if the arguments being bound are the same.+
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+
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   An alpha_ty for a constructor is true if there exist permutations+
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   as above such that a conjunction of propositions for all arguments holds.+
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+
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   For an argument we allow bindings where only one of the following+
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   holds:+
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+
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   - Argument is bound in some shallow bindings: We return true+
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   - Argument of type ty is bound recursively in some other+
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     arguments [i1, .. in] with one binding function bn.+
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     We return:+
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+
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     (bn argl, (argl, argl_i1, ..., argl_in)) \<approx>gen+
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     \<lambda>(argl,argl1,..,argln) (argr,argr1,..,argrn). +
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         (alpha_ty argl argr) \<and> (alpha_i1 argl1 argr1) \<and> .. \<and> (alpha_in argln argrn)+
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     \<lambda>(arg,arg1,..,argn). (fv_ty arg) \<union> (fv_i1 arg1) \<union> .. \<union> (fv_in argn)+
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     pi+
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     (bn argr, (argr, argr_i1, ..., argr_in))+
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+
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   - Argument is bound in some deep non-recursive bindings.+
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     We return: alpha_bn argl argr+
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   - Argument of type ty has some shallow bindings [b1..bn] and/or+
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     non-recursive bindings [f1 a1, .., fm am], where the bindings+
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     have the permutations p1..pl. We return:+
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+
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     (b1l \<union>..\<union> bnl \<union> f1 a1l \<union>..\<union> fn anl, argl) \<approx>gen+
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     alpha_ty fv_ty (p1 +..+ pl)+
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     (b1r \<union>..\<union> bnr \<union> f1 a1r \<union>..\<union> fn anr, argr)+
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+
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   - Argument has some recursive bindings. The bindings were+
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     already treated in 2nd case so we return: True+
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   - Argument has no bindings and is not bound.+
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     If it is recursive for type ty, we return: alpha_ty argl argr+
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     Otherwise we return: argl = argr+
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+
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*)+
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+
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+
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ML {*+
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datatype alpha_mode = AlphaGen | AlphaRes | AlphaLst;+
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*}+
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+
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ML {*+
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fun atyp_const AlphaGen = @{const_name alpha_gen}+
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  | atyp_const AlphaRes = @{const_name alpha_res}+
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  | atyp_const AlphaLst = @{const_name alpha_lst}+
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*}+
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+
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(* TODO: make sure that parser checks that bindings are compatible *)+
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ML {*+
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fun alpha_const_for_binds [] = atyp_const AlphaGen+
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  | alpha_const_for_binds ((NONE, _, _, at) :: t) = atyp_const at+
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  | alpha_const_for_binds ((SOME (_, _), _, _, at) :: _) = atyp_const at+
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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 _ _ = false;+
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+
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fun is_atom_fset thy (Type ("FSet.fset", [t])) = is_atom thy t+
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  | is_atom_fset _ _ = false;+
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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, _, aty) => (f, bi, aty)) binds+
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      val nodups = distinct (op =) common+
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      fun find_bodys (sf, sbi, sty) =+
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        filter (fn (f, bi, _, aty) => f = sf andalso bi = sbi andalso aty = sty) 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, aty), bos), pi) => map (fn bo => ((f, bi, bo, aty), 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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*}+
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ML {*+
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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_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_atom_set 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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  fun mk_atom_fset t =+
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    let+
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      val ty = fastype_of t;+
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      val atom_ty = dest_fsetT ty --> @{typ atom};+
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      val fmap_ty = atom_ty --> ty --> @{typ "atom fset"};+
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      val fset_to_set = @{term "fset_to_set :: atom fset \<Rightarrow> atom set"}+
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    in+
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      fset_to_set $ ((Const (@{const_name fmap}, fmap_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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(* Given [fv1, fv2, fv3] creates %(x, y, z). fv1 x u fv2 y u fv3 z *)+
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ML {*+
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fun mk_compound_fv fvs =+
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let+
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  val nos = (length fvs - 1) downto 0;+
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  val fvs_applied = map (fn (fv, no) => fv $ Bound no) (fvs ~~ nos);+
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  val fvs_union = mk_union fvs_applied;+
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  val (tyh :: tys) = rev (map (domain_type o fastype_of) fvs);+
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  fun fold_fun ty t = HOLogic.mk_split (Abs ("", ty, t))+
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in+
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  fold fold_fun tys (Abs ("", tyh, fvs_union))+
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end;+
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*}+
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+
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(* Given [R1, R2, R3] creates %(x,x'). %(y,y'). %(z,z'). R x x' \<and> R y y' \<and> R z z' *)+
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ML {*+
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fun mk_compound_alpha Rs =+
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let+
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  val nos = (length Rs - 1) downto 0;+
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  val nos2 = (2 * length Rs - 1) downto length Rs;+
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  val Rs_applied = map (fn (R, (no2, no)) => R $ Bound no2 $ Bound no) (Rs ~~ (nos2 ~~ nos));+
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  val Rs_conj = mk_conjl Rs_applied;+
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  val (tyh :: tys) = rev (map (domain_type o fastype_of) Rs);+
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  fun fold_fun ty t = HOLogic.mk_split (Abs ("", ty, t))+
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  val abs_rhs = fold fold_fun tys (Abs ("", tyh, Rs_conj))+
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in+
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  fold fold_fun tys (Abs ("", tyh, abs_rhs))+
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end;+
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*}+
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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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(* 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_fvbn (fvbn, (bn, ith_dtyp, args_in_bns)) =+
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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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  fun fv_bn_constr (cname, dts) args_in_bn =+
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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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(*        val _ = tracing ("B 1" ^ PolyML.makestring args_in_bn);*)+
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(*        val _ = tracing ("B 2" ^ PolyML.makestring bn_fvbn);*)+
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      in+
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        case AList.lookup (op=) args_in_bn arg_no of+
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          SOME NONE => @{term "{} :: atom set"}+
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        | SOME (SOME (f : term)) => (the (AList.lookup (op=) bn_fvbn f)) $ x+
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        | NONE =>+
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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_atom_set x else+
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            if is_atom_fset thy ty then mk_atom_fset 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+
−
    HOLogic.mk_Trueprop (HOLogic.mk_eq+
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      (fvbn $ list_comb (c, args), mk_union (map fv_arg (dts ~~ args ~~ arg_nos))))+
−
  end;+
−
  val (_, (_, _, constrs)) = nth descr ith_dtyp;+
−
  val eqs = map2i fv_bn_constr constrs args_in_bns+
−
in+
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  ((bn, fvbn), eqs)+
−
end+
−
*}+
−
+
−
ML {* print_depth 100 *}+
−
ML {*+
−
fun fv_bns thy dt_info fv_frees rel_bns =+
−
let+
−
  fun mk_fvbn_free (bn, ith, _) =+
−
    let+
−
      val fvbn_name = "fv_" ^ (Long_Name.base_name (fst (dest_Const bn)));+
−
    in+
−
      (fvbn_name, Free (fvbn_name, fastype_of (nth fv_frees ith)))+
−
    end;+
−
  val (fvbn_names, fvbn_frees) = split_list (map mk_fvbn_free rel_bns);+
−
  val bn_fvbn = (map (fn (bn, _, _) => bn) rel_bns) ~~ fvbn_frees+
−
  val (l1, l2) = split_list (map (fv_bn thy dt_info fv_frees bn_fvbn) (fvbn_frees ~~ rel_bns));+
−
in+
−
  (l1, (fvbn_names ~~ l2))+
−
end+
−
*}+
−
+
−
+
−
ML {*+
−
fun alpha_bn (dt_info : Datatype_Aux.info) alpha_frees bn_alphabn ((bn, ith_dtyp, args_in_bns), (alpha_bn_free, _ (*is_rec*) )) =+
−
let+
−
  val {descr, sorts, ...} = dt_info;+
−
  fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);+
−
  fun alpha_bn_constr (cname, dts) args_in_bn =+
−
  let+
−
    val Ts = map (typ_of_dtyp descr sorts) dts;+
−
    val names = Name.variant_list ["pi"] (Datatype_Prop.make_tnames Ts);+
−
    val names2 = Name.variant_list ("pi" :: names) (Datatype_Prop.make_tnames Ts);+
−
    val args = map Free (names ~~ Ts);+
−
    val args2 = map Free (names2 ~~ Ts);+
−
    val c = Const (cname, Ts ---> (nth_dtyp ith_dtyp));+
−
    val rhs = HOLogic.mk_Trueprop+
−
      (alpha_bn_free $ (list_comb (c, args)) $ (list_comb (c, args2)));+
−
    fun lhs_arg ((dt, arg_no), (arg, arg2)) =+
−
      case AList.lookup (op=) args_in_bn arg_no of+
−
        SOME NONE => @{term True}+
−
      | SOME (SOME f) => (the (AList.lookup (op=) bn_alphabn f)) $ arg $ arg2+
−
      | NONE =>+
−
          if is_rec_type dt then (nth alpha_frees (body_index dt)) $ arg $ arg2+
−
          else HOLogic.mk_eq (arg, arg2)+
−
    val arg_nos = 0 upto (length dts - 1)+
−
    val lhss = mk_conjl (map lhs_arg (dts ~~ arg_nos ~~ (args ~~ args2)))+
−
    val eq = Logic.mk_implies (HOLogic.mk_Trueprop lhss, rhs)+
−
  in+
−
    eq+
−
  end+
−
  val (_, (_, _, constrs)) = nth descr ith_dtyp;+
−
  val eqs = map2i alpha_bn_constr constrs args_in_bns+
−
in+
−
  ((bn, alpha_bn_free), eqs)+
−
end+
−
*}+
−
+
−
ML {*+
−
fun alpha_bns dt_info alpha_frees rel_bns bns_rec =+
−
let+
−
  val {descr, sorts, ...} = dt_info;+
−
  fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);+
−
  fun mk_alphabn_free (bn, ith, _) =+
−
    let+
−
      val alphabn_name = "alpha_" ^ (Long_Name.base_name (fst (dest_Const bn)));+
−
      val alphabn_type = nth_dtyp ith --> nth_dtyp ith --> @{typ bool};+
−
      val alphabn_free = Free(alphabn_name, alphabn_type);+
−
    in+
−
      (alphabn_name, alphabn_free)+
−
    end;+
−
  val (alphabn_names, alphabn_frees) = split_list (map mk_alphabn_free rel_bns);+
−
  val bn_alphabn = (map (fn (bn, _, _) => bn) rel_bns) ~~ alphabn_frees;+
−
  val pair = split_list (map (alpha_bn dt_info alpha_frees bn_alphabn)+
−
    (rel_bns ~~ (alphabn_frees ~~ bns_rec)))+
−
in+
−
  (alphabn_names, pair)+
−
end+
−
*}+
−
+
−
+
−
(* Checks that a list of bindings contains only compatible ones *)+
−
ML {*+
−
fun bns_same l =+
−
  length (distinct (op =) (map (fn ((b, _, _, atyp), _) => (b, atyp)) l)) = 1+
−
*}+
−
+
−
ML {*+
−
fun setify x =+
−
  if fastype_of x = @{typ "atom list"} then+
−
  Const (@{const_name set}, @{typ "atom list \<Rightarrow> atom set"}) $ x else x+
−
*}+
−
+
−
ML {*+
−
fun define_fv (dt_info : Datatype_Aux.info) bindsall bns lthy =+
−
let+
−
  val thy = ProofContext.theory_of lthy;+
−
  val {descr, sorts, ...} = dt_info;+
−
  fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);+
−
  val fv_names = Datatype_Prop.indexify_names (map (fn (i, _) =>+
−
    "fv_" ^ name_of_typ (nth_dtyp i)) descr);+
−
  val fv_types = map (fn (i, _) => nth_dtyp i --> @{typ "atom set"}) descr;+
−
  val fv_frees = map Free (fv_names ~~ fv_types);+
−
(* TODO: We need a transitive closure, but instead we do this hack considering+
−
   all binding functions as recursive or not *)+
−
  val nr_bns =+
−
    if (non_rec_binds bindsall) = [] then []+
−
    else map (fn (bn, _, _) => bn) bns;+
−
  val rel_bns = filter (fn (bn, _, _) => bn mem nr_bns) bns;+
−
  val (bn_fv_bns, fv_bn_names_eqs) = fv_bns thy dt_info fv_frees rel_bns;+
−
  val fvbns = map snd bn_fv_bns;+
−
  val (fv_bn_names, fv_bn_eqs) = split_list fv_bn_names_eqs;+
−
+
−
  fun fv_constr ith_dtyp (cname, dts) bindcs =+
−
    let+
−
      val Ts = map (typ_of_dtyp descr sorts) dts;+
−
      val bindslen = length bindcs+
−
      val pi_strs_same = replicate bindslen "pi"+
−
      val pi_strs = Name.variant_list [] pi_strs_same;+
−
      val pis = map (fn ps => Free (ps, @{typ perm})) pi_strs;+
−
      val bind_pis_gath = bindcs ~~ pis;+
−
      val bind_pis = un_gather_binds_cons bind_pis_gath;+
−
      val bindcs = map fst bind_pis;+
−
      val names = Name.variant_list pi_strs (Datatype_Prop.make_tnames Ts);+
−
      val args = map Free (names ~~ Ts);+
−
      val c = Const (cname, Ts ---> (nth_dtyp ith_dtyp));+
−
      val fv_c = nth fv_frees ith_dtyp;+
−
      val arg_nos = 0 upto (length dts - 1)+
−
      fun fv_bind args (NONE, i, _, _) =+
−
            if is_rec_type (nth dts i) then (nth fv_frees (body_index (nth dts i))) $ (nth args i) else+
−
            if ((is_atom thy) o fastype_of) (nth args i) then mk_single_atom (nth args i) else+
−
            if ((is_atom_set thy) o fastype_of) (nth args i) then mk_atom_set (nth args i) else+
−
            if ((is_atom_fset thy) o fastype_of) (nth args i) then mk_atom_fset (nth args i) else+
−
            (* TODO goes the code for preiously defined nominal datatypes *)+
−
            @{term "{} :: atom set"}+
−
        | fv_bind args (SOME (f, _), i, _, _) = f $ (nth args i)+
−
      fun fv_binds_as_set args relevant = mk_union (map (setify o fv_bind args) relevant)+
−
      fun find_nonrec_binder j (SOME (f, false), i, _, _) = if i = j then SOME f else NONE+
−
        | find_nonrec_binder _ _ = NONE+
−
      fun fv_arg ((dt, x), arg_no) =+
−
        case get_first (find_nonrec_binder arg_no) bindcs of+
−
          SOME f =>+
−
            (case get_first (fn (x, y) => if x = f then SOME y else NONE) bn_fv_bns of+
−
                SOME fv_bn => fv_bn $ x+
−
              | NONE => error "bn specified in a non-rec binding but not in bn list")+
−
        | NONE =>+
−
            let+
−
              val arg =+
−
                if is_rec_type dt then nth fv_frees (body_index dt) $ x else+
−
                if ((is_atom thy) o fastype_of) x then mk_single_atom x else+
−
                if ((is_atom_set thy) o fastype_of) x then mk_atom_set x else+
−
                if ((is_atom_fset thy) o fastype_of) x then mk_atom_fset x else+
−
                (* TODO goes the code for preiously defined nominal datatypes *)+
−
                @{term "{} :: atom set"};+
−
              (* If i = j then we generate it only once *)+
−
              val relevant = filter (fn (_, i, j, _) => ((i = arg_no) orelse (j = arg_no))) bindcs;+
−
              val sub = fv_binds_as_set args relevant+
−
            in+
−
              mk_diff arg sub+
−
            end;+
−
      val fv_eq = HOLogic.mk_Trueprop (HOLogic.mk_eq+
−
        (fv_c $ list_comb (c, args), mk_union (map fv_arg  (dts ~~ args ~~ arg_nos))))+
−
    in+
−
      fv_eq+
−
    end;+
−
  fun fv_eq (i, (_, _, constrs)) binds = map2i (fv_constr i) constrs binds;+
−
  val fveqs = map2i fv_eq descr (gather_binds bindsall)+
−
  val fv_eqs_perfv = fveqs+
−
  val rel_bns_nos = map (fn (_, i, _) => i) rel_bns;+
−
  fun filter_fun (_, b) = b mem rel_bns_nos;+
−
  val all_fvs = (fv_names ~~ fv_eqs_perfv) ~~ (0 upto (length fv_names - 1))+
−
  val (fv_names_fst, fv_eqs_fst) = apsnd flat (split_list (map fst (filter_out filter_fun all_fvs)))+
−
  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;+
−
  val add_binds = map (fn x => (Attrib.empty_binding, x))+
−
(* Function_Fun.add_fun Function_Common.default_config ... true *)+
−
  val (fvs, lthy') = (Primrec.add_primrec+
−
    (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+
−
    (map (fn s => (Binding.name s, NONE, NoSyn)) fv_names_snd) (add_binds fv_eqs_snd) lthy')+
−
  val ordered_fvs = fv_frees @ fvbns;+
−
  val all_fvs = (fst fvs @ fst fvs2, snd fvs @ snd fvs2)+
−
in+
−
  ((all_fvs, ordered_fvs), lthy'')+
−
end+
−
*}+
−
+
−
ML {*+
−
fun define_alpha (dt_info : Datatype_Aux.info) bindsall bns fv_frees lthy =+
−
let+
−
  val thy = ProofContext.theory_of lthy;+
−
  val {descr, sorts, ...} = dt_info;+
−
  fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);+
−
(* TODO: We need a transitive closure, but instead we do this hack considering+
−
   all binding functions as recursive or not *)+
−
  val nr_bns =+
−
    if (non_rec_binds bindsall) = [] then []+
−
    else map (fn (bn, _, _) => bn) bns;+
−
  val alpha_names = Datatype_Prop.indexify_names (map (fn (i, _) =>+
−
    "alpha_" ^ name_of_typ (nth_dtyp i)) descr);+
−
  val alpha_types = map (fn (i, _) => nth_dtyp i --> nth_dtyp i --> @{typ bool}) descr;+
−
  val alpha_frees = map Free (alpha_names ~~ alpha_types);+
−
  (* We assume that a bn is either recursive or not *)+
−
  val bns_rec = map (fn (bn, _, _) => not (bn mem nr_bns)) bns;+
−
  val (alpha_bn_names, (bn_alpha_bns, alpha_bn_eqs)) =+
−
    alpha_bns dt_info alpha_frees bns bns_rec+
−
  val alpha_bn_frees = map snd bn_alpha_bns;+
−
  val alpha_bn_types = map fastype_of alpha_bn_frees;+
−
+
−
  fun alpha_constr ith_dtyp (cname, dts) bindcs =+
−
    let+
−
      val Ts = map (typ_of_dtyp descr sorts) dts;+
−
      val bindslen = length bindcs+
−
      val pi_strs_same = replicate bindslen "pi"+
−
      val pi_strs = Name.variant_list [] pi_strs_same;+
−
      val pis = map (fn ps => Free (ps, @{typ perm})) pi_strs;+
−
      val bind_pis_gath = bindcs ~~ pis;+
−
      val bind_pis = un_gather_binds_cons bind_pis_gath;+
−
      val names = Name.variant_list pi_strs (Datatype_Prop.make_tnames Ts);+
−
      val args = map Free (names ~~ Ts);+
−
      val names2 = Name.variant_list (pi_strs @ names) (Datatype_Prop.make_tnames Ts);+
−
      val args2 = map Free (names2 ~~ Ts);+
−
      val c = Const (cname, Ts ---> (nth_dtyp ith_dtyp));+
−
      val alpha = nth alpha_frees ith_dtyp;+
−
      val arg_nos = 0 upto (length dts - 1)+
−
      fun fv_bind args (NONE, i, _, _) =+
−
            if is_rec_type (nth dts i) then (nth fv_frees (body_index (nth dts i))) $ (nth args i) else+
−
            if ((is_atom thy) o fastype_of) (nth args i) then mk_single_atom (nth args i) else+
−
            if ((is_atom_set thy) o fastype_of) (nth args i) then mk_atom_set (nth args i) else+
−
            if ((is_atom_fset thy) o fastype_of) (nth args i) then mk_atom_fset (nth args i) else+
−
            (* TODO goes the code for preiously defined nominal datatypes *)+
−
            @{term "{} :: atom set"}+
−
        | fv_bind args (SOME (f, _), i, _, _) = f $ (nth args i)+
−
      fun fv_binds args relevant = mk_union (map (fv_bind args) relevant)+
−
      val alpha_rhs =+
−
        HOLogic.mk_Trueprop (alpha $ (list_comb (c, args)) $ (list_comb (c, args2)));+
−
      fun alpha_arg ((dt, arg_no), (arg, arg2)) =+
−
        let+
−
          val rel_in_simp_binds = filter (fn ((NONE, i, _, _), _) => i = arg_no | _ => false) bind_pis;+
−
          val rel_in_comp_binds = filter (fn ((SOME _, i, _, _), _) => i = arg_no | _ => false) bind_pis;+
−
          val rel_has_binds = filter (fn ((NONE, _, j, _), _) => j = arg_no+
−
                                       | ((SOME (_, false), _, j, _), _) => j = arg_no+
−
                                       | _ => false) bind_pis;+
−
          val rel_has_rec_binds = filter+
−
            (fn ((SOME (_, true), _, j, _), _) => j = arg_no | _ => false) bind_pis;+
−
        in+
−
          case (rel_in_simp_binds, rel_in_comp_binds, rel_has_binds, rel_has_rec_binds) of+
−
            ([], [], [], []) =>+
−
              if is_rec_type dt then (nth alpha_frees (body_index dt) $ arg $ arg2)+
−
              else (HOLogic.mk_eq (arg, arg2))+
−
          | (_, [], [], []) => @{term True}+
−
          | ([], [], [], _) => @{term True}+
−
          | ([], ((((SOME (bn, is_rec)), _, _, atyp), _) :: _), [], []) =>+
−
            if not (bns_same rel_in_comp_binds) then error "incompatible bindings for an argument" else+
−
            if is_rec then+
−
              let+
−
                val (rbinds, rpis) = split_list rel_in_comp_binds+
−
                val bound_in_nos = map (fn (_, _, i, _) => i) rbinds+
−
                val bound_in_ty_nos = map (fn i => body_index (nth dts i)) bound_in_nos;+
−
                val bound_args = arg :: map (nth args) bound_in_nos;+
−
                val bound_args2 = arg2 :: map (nth args2) bound_in_nos;+
−
                val lhs_binds = fv_binds args rbinds+
−
                val lhs_arg = foldr1 HOLogic.mk_prod bound_args+
−
                val lhs = mk_pair (lhs_binds, lhs_arg);+
−
                val rhs_binds = fv_binds args2 rbinds;+
−
                val rhs_arg = foldr1 HOLogic.mk_prod bound_args2;+
−
                val rhs = mk_pair (rhs_binds, rhs_arg);+
−
                val fvs = map (nth fv_frees) ((body_index dt) :: bound_in_ty_nos);+
−
                val fv = mk_compound_fv fvs;+
−
                val alphas = map (nth alpha_frees) ((body_index dt) :: bound_in_ty_nos);+
−
                val alpha = mk_compound_alpha alphas;+
−
                val pi = foldr1 (uncurry mk_plus) (distinct (op =) rpis);+
−
                val alpha_gen_pre = Const (atyp_const atyp, dummyT) $ lhs $ alpha $ fv $ pi $ rhs;+
−
                val alpha_gen = Syntax.check_term lthy alpha_gen_pre+
−
              in+
−
                alpha_gen+
−
              end+
−
            else+
−
              let+
−
                val alpha_bn_const =+
−
                  nth alpha_bn_frees (find_index (fn (b, _, _) => b = bn) bns)+
−
              in+
−
                alpha_bn_const $ arg $ arg2+
−
              end+
−
          | ([], [], relevant, []) =>+
−
            let+
−
              val (rbinds, rpis) = split_list relevant+
−
              val lhs_binds = fv_binds args rbinds+
−
              val lhs = mk_pair (lhs_binds, arg);+
−
              val rhs_binds = fv_binds args2 rbinds;+
−
              val rhs = mk_pair (rhs_binds, arg2);+
−
              val alpha = nth alpha_frees (body_index dt);+
−
              val fv = nth fv_frees (body_index dt);+
−
              val pi = foldr1 (uncurry mk_plus) (distinct (op =) rpis);+
−
              val alpha_const = alpha_const_for_binds rbinds;+
−
              val alpha_gen_pre = Const (alpha_const, dummyT) $ lhs $ alpha $ fv $ pi $ rhs;+
−
              val alpha_gen = Syntax.check_term lthy alpha_gen_pre+
−
            in+
−
              alpha_gen+
−
            end+
−
          | _ => error "Fv.alpha: not supported binding structure"+
−
        end+
−
      val alphas = map alpha_arg (dts ~~ arg_nos ~~ (args ~~ args2))+
−
      val alpha_lhss = mk_conjl alphas+
−
      val alpha_lhss_ex =+
−
        fold (fn pi_str => fn t => HOLogic.mk_exists (pi_str, @{typ perm}, t)) pi_strs alpha_lhss+
−
      val alpha_eq = Logic.mk_implies (HOLogic.mk_Trueprop alpha_lhss_ex, alpha_rhs)+
−
    in+
−
      alpha_eq+
−
    end;+
−
  fun alpha_eq (i, (_, _, constrs)) binds = map2i (alpha_constr i) constrs binds;+
−
  val alphaeqs = map2i alpha_eq descr (gather_binds bindsall)+
−
  val alpha_eqs = flat alphaeqs+
−
  val add_binds = map (fn x => (Attrib.empty_binding, x))+
−
  val (alphas, lthy') = (Inductive.add_inductive_i+
−
     {quiet_mode = true, verbose = false, alt_name = Binding.empty,+
−
      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_bn_names)+
−
     (alpha_types @ alpha_bn_types)) []+
−
     (add_binds (alpha_eqs @ flat alpha_bn_eqs)) [] lthy)+
−
in+
−
  (alphas, lthy')+
−
end+
−
*}+
−
+
−
+
−
ML {*+
−
fun define_fv_alpha_export dt binds bns ctxt =+
−
let+
−
  val (((fv_ts_loc, fv_def_loc), ord_fv_ts_loc), ctxt') =+
−
    define_fv dt binds bns ctxt;+
−
  val (alpha, ctxt'') =+
−
    define_alpha dt binds bns fv_ts_loc ctxt';+
−
  val alpha_ts_loc = #preds alpha+
−
  val alpha_induct_loc = #induct alpha+
−
  val alpha_intros_loc = #intrs alpha;+
−
  val alpha_cases_loc = #elims alpha+
−
  val morphism = ProofContext.export_morphism ctxt'' ctxt;+
−
  val fv_ts = map (Morphism.term morphism) fv_ts_loc;+
−
  val ord_fv_ts = map (Morphism.term morphism) ord_fv_ts_loc;+
−
  val fv_def = Morphism.fact morphism fv_def_loc;+
−
  val alpha_ts = map (Morphism.term morphism) alpha_ts_loc;+
−
  val alpha_induct = Morphism.thm morphism alpha_induct_loc;+
−
  val alpha_intros = Morphism.fact morphism alpha_intros_loc+
−
  val alpha_cases = Morphism.fact morphism alpha_cases_loc+
−
in+
−
  ((((fv_ts, ord_fv_ts), fv_def), ((alpha_ts, alpha_intros), (alpha_cases, alpha_induct))), ctxt'')+
−
end;+
−
*}+
−
+
−
end+
−