theory Fv

imports "Nominal2_Atoms" "Abs" "Perm" (* For testing *)

begin

(* Bindings are given as a list which has a length being equal

   to the length of the number of constructors.

   Each element is a list whose length is equal to the number

   of arguents.

   Every element specifies bindings of this argument given as

   a tuple: function, bound argument.

  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)]],

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

A SOME binding has to have a function returning an atom set,

and 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.

*)

(* TODO: should be const_name union *)

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

      HOLogic.mk_binop "Set.union" (a, b)) (rev sets) noatoms;

  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;

  (* Copy from Term *)

  fun is_funtype (Type ("fun", [_, _])) = true

    | is_funtype _ = false;

  (* 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 *}

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

ML {*

fun define_fv_alpha (dt_info : Datatype_Aux.info) bindsall 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 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);

  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 = replicate bindslen "pi"

      val pis = map (fn ps => Free (ps, @{typ perm})) pi_strs;

      val bind_pis = bindcs ~~ 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

            (* TODO we assume that all can be 'atomized' *)

            if (is_funtype o fastype_of) (nth args i) then mk_atoms (nth args i) else

            mk_single_atom (nth args i)

        | fv_bind args (SOME f, i, _) = f $ (nth args i);

      fun fv_binds args relevant = mk_union (map (fv_bind args) relevant)

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

        let

          val arg =

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

            (* TODO: we just assume everything can be 'atomized' *)

            if (is_funtype o fastype_of) x then mk_atoms x else

            HOLogic.mk_set @{typ atom} [mk_atom (fastype_of x) $ x];

          (* 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 relevant = filter (fn ((_, i, j), _) => i = arg_no orelse j = arg_no) bind_pis;

        in

          if relevant = [] then (

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

            else (HOLogic.mk_eq (arg, arg2)))

          else

            if is_rec_type dt then let

              (* THE HARD CASE *)

              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);

              (* TODO: check that pis have empty intersection *)

              val pi = foldr1 add_perm rpis;

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

            in

              Syntax.check_term lthy alpha_gen_pre

            (* TODO Add some test that is makes sense *)

            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 =

        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;

  val add_binds = map (fn x => (Attrib.empty_binding, x))

  val (fvs, lthy') = (Primrec.add_primrec

    (map (fn s => (Binding.name s, NONE, NoSyn)) fv_names) (add_binds fv_eqs) lthy)

  val (alphas, lthy'') = (Inductive.add_inductive_i

     {quiet_mode = false, verbose = true, 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_types) []

     (add_binds alpha_eqs) [] lthy')

in

  ((fvs, alphas), lthy'')

end

*}

(* tests

atom_decl name

datatype ty =

  Var "name set"

ML {* Syntax.check_term @{context} (mk_atoms @{term "a :: name set"}) *}

local_setup {* define_fv_alpha "Fv.ty" [[[[]]]] *}

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 =

  rtac induct THEN_ALL_NEW

  asm_full_simp_tac (HOL_ss addsimps inj) THEN_ALL_NEW

(*  TRY o REPEAT_ALL_NEW (CHANGED o rtac conjI) THEN_ALL_NEW*)

  (rtac @{thm exI[of _ "0 :: perm"]} THEN'

   asm_full_simp_tac (HOL_ss addsimps

     @{thms alpha_gen fresh_star_def fresh_zero_perm permute_zero ball_triv}))

*}

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 =

  (((rtac impI THEN' etac induct) ORELSE' rtac induct) THEN_ALL_NEW

  asm_full_simp_tac (HOL_ss addsimps inj) THEN_ALL_NEW

  (etac @{thm exi_neg} THEN' REPEAT o etac conjE THEN'

  (TRY o REPEAT_ALL_NEW (CHANGED o rtac conjI))) THEN_ALL_NEW

  (asm_full_simp_tac HOL_ss) THEN_ALL_NEW

  (etac @{thm alpha_gen_compose_sym} THEN' 

    (asm_full_simp_tac (HOL_ss addsimps (@{thm atom_eqvt} :: eqvt)))))

*}

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 transp_tac ctxt induct alpha_inj term_inj distinct cases eqvt =

  ((rtac impI THEN' etac induct) ORELSE' rtac induct) THEN_ALL_NEW

  (TRY o rtac allI THEN' imp_elim_tac cases ctxt) THEN_ALL_NEW

  (

    asm_full_simp_tac (HOL_ss addsimps alpha_inj @ term_inj @ distinct)

    THEN_ALL_NEW (etac @{thm exi_sum} THEN' RANGE [atac]) THEN_ALL_NEW

    (REPEAT o etac conjE THEN' (TRY o REPEAT_ALL_NEW (CHANGED o rtac conjI)))

    THEN_ALL_NEW (asm_full_simp_tac HOL_ss) THEN_ALL_NEW

    (etac @{thm alpha_gen_compose_trans} THEN' RANGE[atac]) THEN_ALL_NEW

    (asm_full_simp_tac (HOL_ss addsimps (@{thm atom_eqvt} :: eqvt)))

  )

*}

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 1;

  fun symp_tac' _ = symp_tac alpha_induct alpha_inj eqvt 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*)

end