theory Fv

imports "Nominal2_Atoms" "Abs" "Perm" "Rsp"

begin

(* Bindings are a list of lists of lists of triples.

   The first list represents the datatypes defined.

   The second list represents the constructors.

   The internal list is a list of all the bndings that

   concern the constructor.

   Every triple consists of a function, the binding and

   the body.

  Eg:

nominal_datatype

   C1

 | C2 x y z bind x in z

 | C3 x y z bind f x in z bind g y in z

yields:

[

 [],

 [(NONE, 0, 2)],

 [(SOME (Const f), 0, 2), (Some (Const g), 1, 2)]]

A SOME binding has to have a function which takes an appropriate

argument and returns an atom set. A NONE binding has to be on an

argument that is an atom or an atom set.

How the procedure works:

  For each of the defined datatypes,

  For each of the constructors,

  It creates a union of free variables for each argument.

  For an argument the free variables are the variables minus

  bound variables.

  The variables are:

    For an atom, a singleton set with the atom itself.

    For an atom set, the atom set itself.

    For a recursive argument, the appropriate fv function applied to it.

    (* TODO: This one is not implemented *)

    For other arguments it should be an appropriate fv function stored

      in the database.

  The bound variables are a union of results of all bindings that

  involve the given argument. For a paricular binding the result is:

    For a function applied to an argument this function with the argument.

    For an atom, a singleton set with the atom itself.

    For an atom set, the atom set itself.

    For a recursive argument, the appropriate fv function applied to it.

    (* TODO: This one is not implemented *)

    For other arguments it should be an appropriate fv function stored

      in the database.

  For every argument that is a binding, we add a the same binding in its

  body.

*)

ML {*

fun is_atom thy typ =

  Sign.of_sort thy (typ, @{sort at})

fun is_atom_set thy (Type ("fun", [t, @{typ bool}])) = is_atom thy t

  | is_atom_set thy _ = false;

*}

(* Like map2, only if the second list is empty passes empty lists insted of error *)

ML {*

fun map2i _ [] [] = []

  | map2i f (x :: xs) (y :: ys) = f x y :: map2i f xs ys

  | map2i f (x :: xs) [] = f x [] :: map2i f xs []

  | map2i _ _ _ = raise UnequalLengths;

*}

(* Finds bindings with the same function and binding, and gathers all

   bodys for such pairs

 *)

ML {*

fun gather_binds binds =

let

  fun gather_binds_cons binds =

    let

      val common = map (fn (f, bi, _) => (f, bi)) binds

      val nodups = distinct (op =) common

      fun find_bodys (sf, sbi) =

        filter (fn (f, bi, _) => f = sf andalso bi = sbi) binds

      val bodys = map ((map (fn (_, _, bo) => bo)) o find_bodys) nodups

    in

      nodups ~~ bodys

    end

in

  map (map gather_binds_cons) binds

end

*}

ML {*

fun un_gather_binds_cons binds =

  flat (map (fn (((f, bi), bos), pi) => map (fn bo => ((f, bi, bo), pi)) bos) binds)

*}

ML {*

  open Datatype_Aux; (* typ_of_dtyp, DtRec, ... *);

  (* TODO: It is the same as one in 'nominal_atoms' *)

  fun mk_atom ty = Const (@{const_name atom}, ty --> @{typ atom});

  val noatoms = @{term "{} :: atom set"};

  fun mk_single_atom x = HOLogic.mk_set @{typ atom} [mk_atom (type_of x) $ x];

  fun mk_union sets =

    fold (fn a => fn b =>

      if a = noatoms then b else

      if b = noatoms then a else

      if a = b then a else

      HOLogic.mk_binop @{const_name sup} (a, b)) (rev sets) noatoms;

  val mk_inter = foldr1 (HOLogic.mk_binop @{const_name inf})

  fun mk_conjl props =

    fold (fn a => fn b =>

      if a = @{term True} then b else

      if b = @{term True} then a else

      HOLogic.mk_conj (a, b)) props @{term True};

  fun mk_diff a b =

    if b = noatoms then a else

    if b = a then noatoms else

    HOLogic.mk_binop @{const_name minus} (a, b);

  fun mk_atoms t =

    let

      val ty = fastype_of t;

      val atom_ty = HOLogic.dest_setT ty --> @{typ atom};

      val img_ty = atom_ty --> ty --> @{typ "atom set"};

    in

      (Const (@{const_name image}, img_ty) $ Const (@{const_name atom}, atom_ty) $ t)

    end;

  (* Similar to one in USyntax *)

  fun mk_pair (fst, snd) =

    let val ty1 = fastype_of fst

      val ty2 = fastype_of snd

      val c = HOLogic.pair_const ty1 ty2

    in c $ fst $ snd

    end;

*}

ML {* fun add_perm (p1, p2) = Const(@{const_name plus}, @{typ "perm \<Rightarrow> perm \<Rightarrow> perm"}) $ p1 $ p2 *}

ML {*

fun non_rec_binds l =

let

  fun is_non_rec (SOME (f, false), _, _) = SOME f

    | is_non_rec _ = NONE

in

  distinct (op =) (map_filter is_non_rec (flat (flat l)))

end

*}

(* We assume no bindings in the type on which bn is defined *)

ML {*

fun fv_bn thy (dt_info : Datatype_Aux.info) fv_frees (bn, ith_dtyp, args_in_bns) =

let

  val {descr, sorts, ...} = dt_info;

  fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);

  val fvbn_name = "fv_" ^ (Long_Name.base_name (fst (dest_Const bn)));

  val fvbn = Free (fvbn_name, fastype_of (nth fv_frees ith_dtyp));

  fun fv_bn_constr (cname, dts) args_in_bn =

  let

    val Ts = map (typ_of_dtyp descr sorts) dts;

    val names = Datatype_Prop.make_tnames Ts;

    val args = map Free (names ~~ Ts);

    val c = Const (cname, Ts ---> (nth_dtyp ith_dtyp));

    fun fv_arg ((dt, x), arg_no) =

      let

        val ty = fastype_of x

      in

        if arg_no mem args_in_bn then 

          (if is_rec_type dt then

            (if body_index dt = ith_dtyp then fvbn $ x else error "fv_bn: not good")

          else @{term "{} :: atom set"}) else

        if is_atom thy ty then mk_single_atom x else

        if is_atom_set thy ty then mk_atoms x else

        if is_rec_type dt then nth fv_frees (body_index dt) $ x else

        @{term "{} :: atom set"}

      end;

    val arg_nos = 0 upto (length dts - 1)

  in

    HOLogic.mk_Trueprop (HOLogic.mk_eq

      (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

  ((bn, fvbn), (fvbn_name, eqs))

end

*}

ML {*

fun alpha_bn thy (dt_info : Datatype_Aux.info) alpha_frees ((bn, ith_dtyp, args_in_bns), is_rec) =

let

  val {descr, sorts, ...} = dt_info;

  fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);

  val alpha_bn_name = "alpha_" ^ (Long_Name.base_name (fst (dest_Const bn)));

  val alpha_bn_type = @{typ perm} --> nth_dtyp ith_dtyp --> nth_dtyp ith_dtyp --> @{typ bool}

  val alpha_bn_free = Free(alpha_bn_name, alpha_bn_type);

  fun alpha_bn_constr (cname, dts) args_in_bn =

  let

    val Ts = map (typ_of_dtyp descr sorts) dts;

    val pi = Free("pi", @{typ perm})

    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 $ pi $ (list_comb (c, args)) $ (list_comb (c, args2)));

    fun lhs_arg ((dt, arg_no), (arg, arg2)) =

      let

        val argty = fastype_of arg;

        val permute = Const (@{const_name permute}, @{typ perm} --> argty --> argty);

        val permarg = if is_rec then permute $ pi $ arg else arg

      in

      if is_rec_type dt then

        if arg_no mem args_in_bn then alpha_bn_free $ pi $ arg $ arg2

        else (nth alpha_frees (body_index dt)) $ permarg $ arg2

      else

        if arg_no mem args_in_bn then @{term True}

        else HOLogic.mk_eq (permarg, arg2)

      end

    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), (alpha_bn_name, eqs))

end

*}

ML {* fn x => split_list(flat x) *}

ML {* fn x => apsnd flat (split_list (map split_list x)) *}

(* TODO: Notice datatypes without bindings and replace alpha with equality *)

ML {*

fun define_fv_alpha (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);

  val nr_bns = non_rec_binds bindsall;

  val rel_bns = filter (fn (bn, _, _) => bn mem nr_bns) bns;

  val (bn_fv_bns, fv_bn_names_eqs) = split_list (map (fv_bn thy dt_info fv_frees) rel_bns);

  val (fv_bn_names, fv_bn_eqs) = split_list fv_bn_names_eqs;

  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 (bn_alpha_bns, alpha_bn_names_eqs) = split_list (map (alpha_bn thy dt_info alpha_frees) (bns ~~ bns_rec))

  val (alpha_bn_names, alpha_bn_eqs) = split_list alpha_bn_names_eqs;

  val alpha_bn_frees = map snd bn_alpha_bns;

  val alpha_bn_types = map fastype_of alpha_bn_frees;

  fun fv_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 bindcs = map fst bind_pis;

      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 fv_c = nth fv_frees 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_atoms (nth args i) else

            (* TODO we do not know what to do with non-atomizable things *)

            @{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)

      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_atoms x else

                (* TODO we do not know what to do with non-atomizable things *)

                @{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 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))))

      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 ((SOME _, _, j), _) => j = arg_no | _ => false) bind_pis;

        in

          case (rel_in_simp_binds, rel_in_comp_binds, rel_has_binds) of

            ([], [], []) =>

              if is_rec_type dt then (nth alpha_frees (body_index dt) $ arg $ arg2)

              else (HOLogic.mk_eq (arg, arg2))

            (* TODO: if more then check and accept *)

          | (_, [], []) => @{term True}

          | ([], [(((SOME (bn, _)), _, _), pi)], []) =>

            let

              val alpha_bn = nth alpha_bn_frees (find_index (fn (b, _, _) => b = bn) bns) $ pi $ arg $ arg2

              val ty = fastype_of (bn $ arg)

              val permute = Const(@{const_name permute}, @{typ perm} --> ty --> ty)

            in

              HOLogic.mk_conj (alpha_bn, HOLogic.mk_eq (permute $ pi $ (bn $ arg), (bn $ 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 add_perm (distinct (op =) rpis);

              val alpha_gen_pre = Const (@{const_name alpha_gen}, dummyT) $ lhs $ alpha $ fv $ pi $ rhs;

              val alpha_gen = Syntax.check_term lthy alpha_gen_pre

(*              val pi_supps = map ((curry op $) @{term "supp :: perm \<Rightarrow> atom set"}) rpis;

              val pi_supps_eq = HOLogic.mk_eq (mk_inter pi_supps, @{term "{} :: atom set"}) *)

            in

(*              if length pi_supps > 1 then HOLogic.mk_conj (alpha_gen, pi_supps_eq) else*)

              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

      (fv_eq, alpha_eq)

    end;

  fun fv_alpha_eq (i, (_, _, constrs)) binds = map2i (fv_alpha_constr i) constrs binds;

  val fveqs_alphaeqs = map2i fv_alpha_eq descr (gather_binds bindsall)

  val (fv_eqs_perfv, alpha_eqs) = apsnd flat (split_list (map split_list fveqs_alphaeqs))

  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 _ = map tracing (map (Syntax.string_of_term @{context}) fv_eqs_all)

  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 (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'')

  val all_fvs = (fst fvs @ fst fvs2, snd fvs @ snd fvs2)

in

  ((all_fvs, alphas), lthy''')

end

*}

(*

atom_decl name

datatype lam =

  VAR "name"

| APP "lam" "lam"

| 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}, true), 0, 1)]], [[]]] [(@{term bi}, 1, [[0]])] *}

print_theorems

*)

(*atom_decl name

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"

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 (Datatype.the_info @{theory} "Fv.rtrm1") 2 *}

local_setup {*

  snd o define_fv_alpha (Datatype.the_info @{theory} "Fv.rtrm1")

  [[[], [], [(NONE, 0, 1)], [(SOME (@{term bv1}, false), 0, 2)]],

  [[], [], []]] [(@{term bv1}, 1, [[], [0], [0, 1]])] *}

print_theorems

*)

(*

atom_decl name

datatype rtrm5 =

  rVr5 "name"

| rLt5 "rlts" "rtrm5" --"bind (bv5 lts) in (rtrm5)"

and rlts =

  rLnil

| rLcons "name" "rtrm5" "rlts"

primrec

  rbv5

where

  "rbv5 rLnil = {}"

| "rbv5 (rLcons n t ltl) = {atom n} \<union> (rbv5 ltl)"

setup {* snd o define_raw_perms (Datatype.the_info @{theory} "Fv.rtrm5") 2 *}

local_setup {* snd o define_fv_alpha (Datatype.the_info @{theory} "Fv.rtrm5")

  [[[], [(SOME (@{term rbv5}, false), 0, 1)]], [[], []]] [(@{term rbv5}, 1, [[], [0, 2]])] *}

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;