theory Nominal2

imports 

  Nominal2_Base Nominal2_Eqvt Nominal2_Abs

uses ("nominal_dt_rawfuns.ML")

     ("nominal_dt_alpha.ML")

     ("nominal_dt_quot.ML")

begin

use "nominal_dt_rawfuns.ML"

ML {* open Nominal_Dt_RawFuns *}

use "nominal_dt_alpha.ML"

ML {* open Nominal_Dt_Alpha *}

use "nominal_dt_quot.ML"

ML {* open Nominal_Dt_Quot *}

text {* TEST *}

ML {*

fun strip_outer_params (Const("all", _) $ Abs (a, T, t)) = strip_outer_params t |>> cons (a, T)

  | strip_outer_params B = ([], B)

fun strip_params_prems_concl trm =

  let

    val (params, body) = strip_outer_params trm

    val (prems, concl) = Logic.strip_horn body

  in

    (params, prems, concl)

  end

fun list_params_prems_concl params prems concl =

  Logic.list_implies (prems, concl)

  |> fold_rev mk_all params

fun mk_binop_env tys c (t, u) =

  let val ty = fastype_of1 (tys, t) in

    Const (c, [ty, ty] ---> ty) $ t $ u

  end

fun mk_union_env tys (t1, @{term "{}::atom set"}) = t1

  | mk_union_env tys (@{term "{}::atom set"}, t2) = t2

  | mk_union_env tys (t1, @{term "set ([]::atom list)"}) = t1

  | mk_union_env tys (@{term "set ([]::atom list)"}, t2) = t2

  | mk_union_env tys (t1, t2) = mk_binop_env tys @{const_name "sup"} (t1, t2)  

fun fold_union_env tys trms = fold_rev (curry (mk_union_env tys)) trms @{term "{}::atom set"}

*}

ML {*

fun process_ecase lthy c (params, prems, concl) bclauses =

  let

    val tys = map snd params

    val binders = get_all_binders bclauses

    fun prep_binder (opt, i) = 

      let

        val t = Bound (length tys - i - 1)

      in

        case opt of

          NONE => setify_ty lthy (nth tys i) t 

        | SOME bn => to_set_ty (fastype_of1 (tys, bn $ t)) (bn $ t)   

      end 

    val fresh_prem = 

      case binders of

          |> map prep_binder

          |> fold_union_env tys

          |> (fn t => mk_fresh_star t c)

          |> HOLogic.mk_Trueprop

          |> single 

  in

    list_params_prems_concl params (fresh_prem @ prems) concl

  end  

*}

ML {*

fun fresh_thm ctxt c parms binders bn_finite_thms =

  let

    fun prep_binder (opt, i) = 

      case opt of

        NONE => setify ctxt (nth parms i) 

      | SOME bn => to_set (bn $ (nth parms i))  

    fun prep_binder2 (opt, i) = 

      case opt of

        NONE => atomify ctxt (nth parms i)

      | SOME bn => bn $ (nth parms i) 

    val rhs = HOLogic.mk_tuple ([c] @ parms @ (map prep_binder2 binders))

    val lhs = binders

      |> map prep_binder

      |> fold_union

      |> mk_perm (Bound 0)

    val goal = mk_fresh_star lhs rhs

      |> (fn t => HOLogic.mk_exists ("p", @{typ perm}, t))

      |> HOLogic.mk_Trueprop   

    val ss = bn_finite_thms @ @{thms supp_Pair finite_supp finite_sets_supp} 

      @ @{thms finite.intros finite_Un finite_set finite_fset}  

  in

    Goal.prove ctxt [] [] goal

      (K (HEADGOAL (rtac @{thm at_set_avoiding1}

          THEN_ALL_NEW (simp_tac (HOL_ss addsimps ss)))))

  end

fun abs_eq_thm ctxt fprops p parms bn_finite_thms (BC (bmode, binders, bodies)) =

  case binders of

    [] => [] 

      let

        val body_trm = foldl1 HOLogic.mk_prod (map (nth parms) bodies)

        val body_ty = fastype_of body_trm

        val (abs_name, binder_ty, abs_ty) = 

          case bmode of

            Lst => (@{const_name "Abs_lst"}, @{typ "atom list"}, @{type_name abs_lst})

          | Set => (@{const_name "Abs_set"}, @{typ "atom set"},  @{type_name abs_set})

          | Res => (@{const_name "Abs_res"}, @{typ "atom set"},  @{type_name abs_res})

        val abs = Const (abs_name, [binder_ty, body_ty] ---> Type (abs_ty, [body_ty]))

        val abs_lhs = abs $ binder_trm $ body_trm

        val abs_rhs = abs $ mk_perm p binder_trm $ Bound 0

        val goal = HOLogic.mk_eq (abs_lhs, abs_rhs)

          |> (fn t => HOLogic.mk_exists ("y", body_ty, t))

          |> HOLogic.mk_Trueprop  

        val ss = fprops @ bn_finite_thms @ @{thms set.simps set_append union_eqvt}

          @ @{thms fresh_star_Un fresh_star_Pair fresh_star_list fresh_star_singleton fresh_star_fset

          fresh_star_set} @ @{thms finite.intros finite_fset}  

     in

       [Goal.prove ctxt [] [] goal

          (K (HEADGOAL (resolve_tac @{thms Abs_rename_set Abs_rename_res Abs_rename_lst}

              THEN_ALL_NEW (simp_tac (HOL_basic_ss addsimps ss) THEN' TRY o simp_tac HOL_ss))))] 

     end

*}

ML {*

fun mk_cperm ctxt p ctrm =

  mk_perm (term_of p) (term_of ctrm)

  |> cterm_of (ProofContext.theory_of ctxt)

*}

ML {*

fun case_tac ctxt c bn_finite_thms (prems, bclausess) thm =

  let

    fun aux_tac prem bclauses =

      case (get_all_binders bclauses) of

        [] => EVERY' [rtac prem, atac]

      | binders => Subgoal.FOCUS (fn {params, prems, context = ctxt, ...} =>  

          let

            val parms = map (term_of o snd) params

            val fthm = fresh_thm ctxt c parms binders bn_finite_thms 

            val ss = @{thms fresh_star_Pair union_eqvt fresh_star_Un}

            val (([(_, fperm)], fprops), ctxt') = Obtain.result

              (K (EVERY1 [etac exE, 

                          full_simp_tac (HOL_basic_ss addsimps ss), 

                          REPEAT o (etac conjE)])) [fthm] ctxt 

            (*

            val _ = tracing ("fprop:\n" ^ cat_lines (map (Syntax.string_of_term ctxt' o prop_of) fprops))

            *)    

            (* 

            val _ = tracing ("abs_eqs:\n" ^ cat_lines (map (Syntax.string_of_term ctxt' o prop_of) abs_eqs))

            *)

          in

            (*HEADGOAL (rtac prem THEN' RANGE [K all_tac, simp_tac (HOL_basic_ss addsimps prems)])*)

            Skip_Proof.cheat_tac (ProofContext.theory_of ctxt')

          end) ctxt

  in

    rtac thm THEN' RANGE (map2 aux_tac prems bclausess)

  end

*}

ML {*

fun prove_strong_exhausts lthy qexhausts qtrms bclausesss bn_finite_thms =

  let

    val ((_, qexhausts'), lthy') = Variable.import true qexhausts lthy 

    val ([c, a], lthy'') = Variable.variant_fixes ["c", "'a"] lthy'

    val c = Free (c, TFree (a, @{sort fs}))

    val (ecases, main_concls) = qexhausts' (* ecases or of the form (params, prems, concl) *)

      |> map prop_of

      |> map Logic.strip_horn

      |> split_list

      |>> (map o map) strip_params_prems_concl     

    val prems = map2 (process_ecase lthy'' c) (flat ecases) (flat bclausesss)

  in

    Goal.prove_multi lthy'' [] prems main_concls

     (fn {prems:thm list, context} =>

        let

           val prems' = partitions prems (map length bclausesss)

        in

          EVERY1 [Goal.conjunction_tac,

            RANGE (map2 (case_tac context c bn_finite_thms) (prems' ~~ bclausesss) qexhausts')]

        end)

  end

*}

ML {*

val eqvt_attr = Attrib.internal (K Nominal_ThmDecls.eqvt_add)

val rsp_attr = Attrib.internal (K Quotient_Info.rsp_rules_add)

val simp_attr = Attrib.internal (K Simplifier.simp_add)

*}

section{* Interface for nominal_datatype *}

ML {* print_depth 50 *}

ML {*

fun get_cnstrs dts =

  map (fn (_, _, _, constrs) => constrs) dts

fun get_typed_cnstrs dts =

  flat (map (fn (_, bn, _, constrs) => 

   (map (fn (bn', _, _) => (Binding.name_of bn, Binding.name_of bn')) constrs)) dts)

fun get_cnstr_strs dts =

  map (fn (bn, _, _) => Binding.name_of bn) (flat (get_cnstrs dts))

fun get_bn_fun_strs bn_funs =

  map (fn (bn_fun, _, _) => Binding.name_of bn_fun) bn_funs

*}

text {* Infrastructure for adding "_raw" to types and terms *}

ML {*

fun add_raw s = s ^ "_raw"

fun add_raws ss = map add_raw ss

fun raw_bind bn = Binding.suffix_name "_raw" bn

fun replace_str ss s = 

  case (AList.lookup (op=) ss s) of 

     SOME s' => s'

   | NONE => s

fun replace_typ ty_ss (Type (a, Ts)) = Type (replace_str ty_ss a, map (replace_typ ty_ss) Ts)

  | replace_typ ty_ss T = T  

fun raw_dts ty_ss dts =

let

  fun raw_dts_aux1 (bind, tys, mx) =

    (raw_bind bind, map (replace_typ ty_ss) tys, mx)

  fun raw_dts_aux2 (ty_args, bind, mx, constrs) =

    (ty_args, raw_bind bind, mx, map raw_dts_aux1 constrs)

in

  map raw_dts_aux2 dts

end

fun replace_aterm trm_ss (Const (a, T)) = Const (replace_str trm_ss a, T)

  | replace_aterm trm_ss (Free (a, T)) = Free (replace_str trm_ss a, T)

  | replace_aterm trm_ss trm = trm

fun replace_term trm_ss ty_ss trm =

  trm |> Term.map_aterms (replace_aterm trm_ss) |> map_types (replace_typ ty_ss) 

*}

ML {*

fun rawify_dts dt_names dts dts_env =

let

  val raw_dts = raw_dts dts_env dts

  val raw_dt_names = add_raws dt_names

in

  (raw_dt_names, raw_dts)

end 

*}

ML {*

fun rawify_bn_funs dts_env cnstrs_env bn_fun_env bn_funs bn_eqs =

let

  val bn_funs' = map (fn (bn, ty, mx) => 

    (raw_bind bn, SOME (replace_typ dts_env ty), mx)) bn_funs

  val bn_eqs' = map (fn (attr, trm) => 

    (attr, replace_term (cnstrs_env @ bn_fun_env) dts_env trm)) bn_eqs

in

  (bn_funs', bn_eqs') 

end 

*}

ML {* 

fun rawify_bclauses dts_env cnstrs_env bn_fun_env bclauses =

let

  fun rawify_bnds bnds = 

    map (apfst (Option.map (replace_term (cnstrs_env @ bn_fun_env) dts_env))) bnds

  fun rawify_bclause (BC (mode, bnds, bdys)) = BC (mode, rawify_bnds bnds, bdys)

in

  (map o map o map) rawify_bclause bclauses

end

*}

ML {*

fun define_raw_dts dts bn_funs bn_eqs bclauses lthy =

let

  val thy = Local_Theory.exit_global lthy

  val thy_name = Context.theory_name thy

  val dt_names = map (fn (_, s, _, _) => Binding.name_of s) dts

  val dt_full_names = map (Long_Name.qualify thy_name) dt_names 

  val dt_full_names' = add_raws dt_full_names

  val dts_env = dt_full_names ~~ dt_full_names'

  val cnstrs = get_cnstr_strs dts

  val cnstrs_ty = get_typed_cnstrs dts

  val cnstrs_full_names = map (Long_Name.qualify thy_name) cnstrs

  val cnstrs_full_names' = map (fn (x, y) => Long_Name.qualify thy_name 

    (Long_Name.qualify (add_raw x) (add_raw y))) cnstrs_ty

  val cnstrs_env = cnstrs_full_names ~~ cnstrs_full_names'

  val bn_fun_strs = get_bn_fun_strs bn_funs

  val bn_fun_strs' = add_raws bn_fun_strs

  val bn_fun_env = bn_fun_strs ~~ bn_fun_strs'

  val bn_fun_full_env = map (pairself (Long_Name.qualify thy_name)) 

    (bn_fun_strs ~~ bn_fun_strs')

  val (raw_dt_names, raw_dts) = rawify_dts dt_names dts dts_env

  val (raw_bn_funs, raw_bn_eqs) = rawify_bn_funs dts_env cnstrs_env bn_fun_env bn_funs bn_eqs 

  val raw_bclauses = rawify_bclauses dts_env cnstrs_env bn_fun_full_env bclauses 

  val (raw_dt_full_names, thy1) = 

    Datatype.add_datatype Datatype.default_config raw_dt_names raw_dts thy

  val lthy1 = Named_Target.theory_init thy1

in

  (raw_dt_full_names, raw_dts, raw_bclauses, raw_bn_funs, raw_bn_eqs, lthy1)

end

*}

ML {*

(* for testing porposes - to exit the procedure early *)

exception TEST of Proof.context

val (STEPS, STEPS_setup) = Attrib.config_int "STEPS" (K 100);

fun get_STEPS ctxt = Config.get ctxt STEPS

*}

setup STEPS_setup

ML {*

fun nominal_datatype2 opt_thms_name dts bn_funs bn_eqs bclauses lthy =

let

  (* definition of the raw datatypes *)

  val _ = warning "Definition of raw datatypes";

  val (raw_dt_names, raw_dts, raw_bclauses, raw_bn_funs, raw_bn_eqs, lthy0) =

    if get_STEPS lthy > 0 

    then define_raw_dts dts bn_funs bn_eqs bclauses lthy

    else raise TEST lthy

  val dtinfo = Datatype.the_info (ProofContext.theory_of lthy0) (hd raw_dt_names)

  val {descr, sorts, ...} = dtinfo

  val raw_tys = all_dtyps descr sorts

  val raw_full_ty_names = map (fst o dest_Type) raw_tys

  val tvs = hd raw_tys

    |> snd o dest_Type

    |> map dest_TFree  

  val dtinfos = map (Datatype.the_info (ProofContext.theory_of lthy0)) raw_full_ty_names  

  val raw_cns_info = all_dtyp_constrs_types descr sorts

  val raw_constrs = (map o map) (fn (c, _, _, _) => c) raw_cns_info

  val raw_inject_thms = flat (map #inject dtinfos)

  val raw_distinct_thms = flat (map #distinct dtinfos)

  val raw_induct_thm = #induct dtinfo

  val raw_induct_thms = #inducts dtinfo

  val raw_exhaust_thms = map #exhaust dtinfos

  val raw_size_trms = map HOLogic.size_const raw_tys

  val raw_size_thms = Size.size_thms (ProofContext.theory_of lthy0) (hd raw_dt_names)

    |> `(fn thms => (length thms) div 2)

    |> uncurry drop

  (* definitions of raw permutations by primitive recursion *)

  val _ = warning "Definition of raw permutations";

  val ((raw_perm_funs, raw_perm_simps, raw_perm_laws), lthy2a) =

    if get_STEPS lthy0 > 0 

    then define_raw_perms raw_full_ty_names raw_tys tvs (flat raw_constrs) raw_induct_thm lthy0

    else raise TEST lthy0

  (* noting the raw permutations as eqvt theorems *)

  val (_, lthy3) = Local_Theory.note ((Binding.empty, [eqvt_attr]), raw_perm_simps) lthy2a

  (* definition of raw fv and bn functions *)

  val _ = warning "Definition of raw fv- and bn-functions";

  val (raw_bns, raw_bn_defs, raw_bn_info, raw_bn_induct, lthy3a) =

    if get_STEPS lthy3 > 1

    then define_raw_bns raw_full_ty_names raw_dts raw_bn_funs raw_bn_eqs 

      (raw_inject_thms @ raw_distinct_thms) raw_size_thms lthy3

    else raise TEST lthy3

  (* defining the permute_bn functions *)

  val (raw_perm_bns, raw_perm_bn_simps, lthy3b) = 

    if get_STEPS lthy3a > 2

    then define_raw_bn_perms raw_tys raw_bn_info raw_cns_info 

      (raw_inject_thms @ raw_distinct_thms) raw_size_thms lthy3a

    else raise TEST lthy3a

  val (raw_fvs, raw_fv_bns, raw_fv_defs, raw_fv_bns_induct, lthy3c) = 

    if get_STEPS lthy3b > 3 

    then define_raw_fvs raw_full_ty_names raw_tys raw_cns_info raw_bn_info raw_bclauses 

      (raw_inject_thms @ raw_distinct_thms) raw_size_thms lthy3b

    else raise TEST lthy3b

  (* definition of raw alphas *)

  val _ = warning "Definition of alphas";

  val (alpha_trms, alpha_bn_trms, alpha_intros, alpha_cases, alpha_induct, lthy4) =

    if get_STEPS lthy3c > 4 

    then define_raw_alpha raw_full_ty_names raw_tys raw_cns_info raw_bn_info raw_bclauses raw_fvs lthy3c

    else raise TEST lthy3c

  val alpha_tys = map (domain_type o fastype_of) alpha_trms  

  (* definition of alpha-distinct lemmas *)

  val _ = warning "Distinct theorems";

  val alpha_distincts = 

    mk_alpha_distincts lthy4 alpha_cases raw_distinct_thms alpha_trms raw_tys

  (* definition of alpha_eq_iff  lemmas *)

  val _ = warning "Eq-iff theorems";

  val alpha_eq_iff = 

    if get_STEPS lthy > 5

    then mk_alpha_eq_iff lthy4 alpha_intros raw_distinct_thms raw_inject_thms alpha_cases

    else raise TEST lthy4

  (* proving equivariance lemmas for bns, fvs, size and alpha *)

  val _ = warning "Proving equivariance";

  val raw_bn_eqvt = 

    if get_STEPS lthy > 6

    then raw_prove_eqvt raw_bns raw_bn_induct (raw_bn_defs @ raw_perm_simps) lthy4

    else raise TEST lthy4

  (* noting the raw_bn_eqvt lemmas in a temprorary theory *)

  val lthy_tmp = snd (Local_Theory.note ((Binding.empty, [eqvt_attr]), raw_bn_eqvt) lthy4)

  val raw_fv_eqvt = 

    if get_STEPS lthy > 7

    then raw_prove_eqvt (raw_fvs @ raw_fv_bns) raw_fv_bns_induct (raw_fv_defs @ raw_perm_simps) 

      (Local_Theory.restore lthy_tmp)

    else raise TEST lthy4

  val raw_size_eqvt = 

    if get_STEPS lthy > 8

    then raw_prove_eqvt raw_size_trms raw_induct_thms (raw_size_thms @ raw_perm_simps) 

      (Local_Theory.restore lthy_tmp)

      |> map (rewrite_rule @{thms permute_nat_def[THEN eq_reflection]})

      |> map (fn thm => thm RS @{thm sym})

    else raise TEST lthy4

  val lthy5 = snd (Local_Theory.note ((Binding.empty, [eqvt_attr]), raw_fv_eqvt) lthy_tmp)

  val (alpha_eqvt, lthy6) =

    if get_STEPS lthy > 9

    then Nominal_Eqvt.equivariance true (alpha_trms @ alpha_bn_trms) alpha_induct alpha_intros lthy5

    else raise TEST lthy4

  (* proving alpha equivalence *)

  val _ = warning "Proving equivalence"

  val alpha_refl_thms = 

    if get_STEPS lthy > 10

    then raw_prove_refl alpha_trms alpha_bn_trms alpha_intros raw_induct_thm lthy6