theory Nominal2

imports 

  Nominal2_Base Nominal2_Abs

uses ("nominal_dt_rawfuns.ML")

     ("nominal_dt_alpha.ML")

     ("nominal_dt_quot.ML")

     ("nominal_induct.ML")

     ("nominal_inductive.ML")

     ("nominal_function_common.ML")

     ("nominal_function_core.ML")

     ("nominal_mutual.ML")

     ("nominal_function.ML")

     ("nominal_dt_data.ML")

begin

use "nominal_dt_data.ML"

ML {* open Nominal_Dt_Data *}

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 *}

(*****************************************)

(* setup for induction principles method *)

use "nominal_induct.ML"

method_setup nominal_induct =

  {* NominalInduct.nominal_induct_method *}

  {* nominal induction *}

(****************************************************)

(* inductive definition involving nominal datatypes *)

use "nominal_inductive.ML" 

(***************************************)

(* forked code of the function package *)

(* for defining nominal functions      *)

use "nominal_function_common.ML"

use "nominal_function_core.ML"

use "nominal_mutual.ML"

use "nominal_function.ML"

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)

val induct_attr = Attrib.internal (K Induct.induct_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, _) =

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

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

    (ty_args, raw_bind bind, NoSyn, 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, _) => 

    (raw_bind bn, SOME (replace_typ dts_env ty), NoSyn)) 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 {*

(* definition of the raw datatype *)

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 cnstr_names = get_cnstr_strs dts

  val cnstr_tys = get_typed_cnstrs dts

  val cnstr_full_names = map (Long_Name.qualify thy_name) cnstr_names

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

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

  val cnstrs_env = cnstr_full_names ~~ cnstr_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, cnstr_names, lthy1)

end

*}

ML {*

fun nominal_datatype2 opt_thms_name dts bn_funs bn_eqs bclauses lthy =

let

  val _ = trace_msg (K "Defining raw datatypes...")

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

    define_raw_dts dts bn_funs bn_eqs bclauses lthy   

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

  val {descr, sorts, ...} = hd dtinfos

  val raw_tys = Datatype_Aux.get_rec_types descr sorts

  val tvs = hd raw_tys

    |> snd o dest_Type

    |> map dest_TFree  

  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 (hd dtinfos)

  val raw_induct_thms = #inducts (hd dtinfos)

  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

  val _ = trace_msg (K "Defining raw permutations...")

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

    define_raw_perms raw_dt_names raw_tys tvs (flat raw_constrs) raw_induct_thm lthy0

  (* noting the raw permutations as eqvt theorems *)

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

  val _ = trace_msg (K "Defining raw fv- and bn-functions...")

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

    define_raw_bns raw_dt_names raw_dts raw_bn_funs raw_bn_eqs 

      (raw_inject_thms @ raw_distinct_thms) raw_size_thms lthy3

  (* defining the permute_bn functions *)

  val (raw_perm_bns, raw_perm_bn_simps, lthy3b) = 

    define_raw_bn_perms raw_tys raw_bn_info raw_cns_info 

      (raw_inject_thms @ raw_distinct_thms) raw_size_thms lthy3a

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

    define_raw_fvs raw_dt_names raw_tys raw_cns_info raw_bn_info raw_bclauses 

      (raw_inject_thms @ raw_distinct_thms) raw_size_thms lthy3b

  val _ = trace_msg (K "Defining alpha relations...")

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

    define_raw_alpha raw_dt_names raw_tys raw_cns_info raw_bn_info raw_bclauses raw_fvs lthy3c

  val alpha_tys = map (domain_type o fastype_of) alpha_trms  

  val _ = trace_msg (K "Proving distinct theorems...")

  val alpha_distincts = 

    mk_alpha_distincts lthy4 alpha_cases raw_distinct_thms alpha_trms raw_dt_names

  val _ = trace_msg (K "Proving eq-iff theorems...")

  val alpha_eq_iff = 

    mk_alpha_eq_iff lthy4 alpha_intros raw_distinct_thms raw_inject_thms alpha_cases

  val _ = trace_msg (K "Proving equivariance of bns, fvs, size and alpha...")

  val raw_bn_eqvt = 

    raw_prove_eqvt raw_bns raw_bn_induct (raw_bn_defs @ raw_perm_simps) 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 = 

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

      (Local_Theory.restore lthy_tmp)

  val raw_size_eqvt = 

    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})

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

  val alpha_eqvt =

    Nominal_Eqvt.raw_equivariance (alpha_trms @ alpha_bn_trms) alpha_induct alpha_intros lthy5

  val alpha_eqvt_norm = map (Nominal_ThmDecls.eqvt_transform lthy5) alpha_eqvt

  val _ = trace_msg (K "Proving equivalence of alpha...")

  val alpha_refl_thms = 

    raw_prove_refl alpha_trms alpha_bn_trms alpha_intros raw_induct_thm lthy5 

  val alpha_sym_thms = 

    raw_prove_sym (alpha_trms @ alpha_bn_trms) alpha_intros alpha_induct alpha_eqvt_norm lthy5 

  val alpha_trans_thms = 

    raw_prove_trans (alpha_trms @ alpha_bn_trms) (raw_distinct_thms @ raw_inject_thms) 

      alpha_intros alpha_induct alpha_cases alpha_eqvt_norm lthy5

  val (alpha_equivp_thms, alpha_bn_equivp_thms) = 

    raw_prove_equivp alpha_trms alpha_bn_trms alpha_refl_thms alpha_sym_thms 

      alpha_trans_thms lthy5

  val _ = trace_msg (K "Proving alpha implies bn...")

  val alpha_bn_imp_thms = 

    raw_prove_bn_imp alpha_trms alpha_bn_trms alpha_intros alpha_induct lthy5 

  val _ = trace_msg (K "Proving respectfulness...")

  val raw_funs_rsp_aux = 

    raw_fv_bn_rsp_aux alpha_trms alpha_bn_trms raw_fvs 

      raw_bns raw_fv_bns alpha_induct (raw_bn_defs @ raw_fv_defs) lthy5

  val raw_funs_rsp = map mk_funs_rsp raw_funs_rsp_aux

  val raw_size_rsp = 

    raw_size_rsp_aux (alpha_trms @ alpha_bn_trms) alpha_induct 

      (raw_size_thms @ raw_size_eqvt) lthy5

      |> map mk_funs_rsp

  val raw_constrs_rsp = 

    raw_constrs_rsp (flat raw_constrs) alpha_trms alpha_intros

      (alpha_bn_imp_thms @ raw_funs_rsp_aux) lthy5 

  val alpha_permute_rsp = map mk_alpha_permute_rsp alpha_eqvt

  val alpha_bn_rsp = 

    raw_alpha_bn_rsp alpha_bn_trms alpha_bn_equivp_thms alpha_bn_imp_thms

  val raw_perm_bn_rsp =

    raw_perm_bn_rsp (alpha_trms @ alpha_bn_trms) raw_perm_bns alpha_induct 

      alpha_intros raw_perm_bn_simps lthy5

  (* noting the quot_respects lemmas *)

  val (_, lthy6) = 

    Local_Theory.note ((Binding.empty, [rsp_attr]),

      raw_constrs_rsp @ raw_funs_rsp @ raw_size_rsp @ alpha_permute_rsp @ 

      alpha_bn_rsp @ raw_perm_bn_rsp) lthy5

  val _ = trace_msg (K "Defining the quotient types...")

  val qty_descr = map (fn (vs, bind, mx, _) => (vs, bind, mx)) dts

  val (qty_infos, lthy7) = 

    define_qtypes qty_descr alpha_tys alpha_trms alpha_equivp_thms lthy6

  val qtys = map #qtyp qty_infos

  val qty_full_names = map (fst o dest_Type) qtys

  val qty_names = map Long_Name.base_name qty_full_names             

  val _ = trace_msg (K "Defining the quotient constants...")

  val qconstrs_descrs =

    (map2 o map2) (fn (b, _, mx) => fn t => (Variable.check_name b, t, mx)) (get_cnstrs dts) raw_constrs

  val qbns_descr =

    map2 (fn (b, _, mx) => fn t => (Variable.check_name b, t, mx)) bn_funs raw_bns

  val qfvs_descr = 

    map2 (fn n => fn t => ("fv_" ^ n, t, NoSyn)) qty_names raw_fvs

  val qfv_bns_descr = 

    map2 (fn (b, _, _) => fn t => ("fv_" ^ Variable.check_name b, t, NoSyn)) bn_funs raw_fv_bns

  val qalpha_bns_descr = 

    map2 (fn (b, _, _) => fn t => ("alpha_" ^ Variable.check_name b, t, NoSyn)) bn_funs  alpha_bn_trms

  val qperm_descr =

    map2 (fn n => fn t => ("permute_" ^ n, Type.legacy_freeze t, NoSyn)) qty_names raw_perm_funs

  val qsize_descr =

    map2 (fn n => fn t => ("size_" ^ n, t, NoSyn)) qty_names raw_size_trms

  val qperm_bn_descr = 

    map2 (fn (b, _, _) => fn t => ("permute_" ^ Variable.check_name b, t, NoSyn)) bn_funs raw_perm_bns

  val ((((((qconstrs_infos, qbns_info), qfvs_info), qfv_bns_info), qalpha_bns_info), qperm_bns_info), 

    lthy8) = 

      lthy7

      |> fold_map (define_qconsts qtys) qconstrs_descrs 

      ||>> define_qconsts qtys qbns_descr 

      ||>> define_qconsts qtys qfvs_descr

      ||>> define_qconsts qtys qfv_bns_descr

      ||>> define_qconsts qtys qalpha_bns_descr

      ||>> define_qconsts qtys qperm_bn_descr

  val lthy9 = 

    define_qperms qtys qty_full_names tvs qperm_descr raw_perm_laws lthy8 

  val lthy9a = 

    define_qsizes qtys qty_full_names tvs qsize_descr lthy9

  val qtrms = (map o map) #qconst qconstrs_infos

  val qbns = map #qconst qbns_info

  val qfvs = map #qconst qfvs_info

  val qfv_bns = map #qconst qfv_bns_info

  val qalpha_bns = map #qconst qalpha_bns_info

  val qperm_bns = map #qconst qperm_bns_info

  val _ = trace_msg (K "Lifting of theorems...")  

  val eq_iff_simps = @{thms alphas permute_prod.simps prod_fv.simps prod_alpha_def prod_rel_def

    prod.cases} 

  val ((((((qdistincts, qeq_iffs), qfv_defs), qbn_defs), qperm_simps), qfv_qbn_eqvts), lthyA) = 

    lthy9a    

    |> lift_thms qtys [] alpha_distincts  

    ||>> lift_thms qtys eq_iff_simps alpha_eq_iff       

    ||>> lift_thms qtys [] raw_fv_defs

    ||>> lift_thms qtys [] raw_bn_defs

    ||>> lift_thms qtys [] raw_perm_simps

    ||>> lift_thms qtys [] (raw_fv_eqvt @ raw_bn_eqvt)

  val ((((((qsize_eqvt, [qinduct]), qexhausts), qsize_simps), qperm_bn_simps), qalpha_refl_thms), lthyB) = 

    lthyA 

    |> lift_thms qtys [] raw_size_eqvt

    ||>> lift_thms qtys [] [raw_induct_thm]

    ||>> lift_thms qtys [] raw_exhaust_thms

    ||>> lift_thms qtys [] raw_size_thms

    ||>> lift_thms qtys [] raw_perm_bn_simps

    ||>> lift_thms qtys [] alpha_refl_thms

  val qinducts = Project_Rule.projections lthyA qinduct

  val _ = trace_msg (K "Proving supp lemmas and fs-instances...")

  val qsupports_thms =

    prove_supports lthyB qperm_simps (flat qtrms)

  (* finite supp lemmas *)

  val qfsupp_thms = prove_fsupp lthyB qtys qinduct qsupports_thms

  (* fs instances *)

  val lthyC = fs_instance qtys qty_full_names tvs qfsupp_thms lthyB

  val _ = trace_msg (K "Proving equality between fv and supp...")

  val qfv_supp_thms = 

    prove_fv_supp qtys (flat qtrms) qfvs qfv_bns qalpha_bns qfv_defs qeq_iffs 

      qperm_simps qfv_qbn_eqvts qinduct (flat raw_bclauses) lthyC

  (* postprocessing of eq and fv theorems *)

  val qeq_iffs' = qeq_iffs

    |> map (simplify (HOL_basic_ss addsimps qfv_supp_thms))

    |> map (simplify (HOL_basic_ss addsimps @{thms prod_fv_supp prod_alpha_eq Abs_eq_iff[symmetric]}))

  val qsupp_constrs = qfv_defs

    |> map (simplify (HOL_basic_ss addsimps (take (length qfvs) qfv_supp_thms)))

  val transform_thm = @{lemma "x = y \<Longrightarrow> a \<notin> x \<longleftrightarrow> a \<notin> y" by simp}

  val transform_thms = 

    [ @{lemma "a \<notin> (S \<union> T) \<longleftrightarrow> a \<notin> S \<and> a \<notin> T" by simp}, 

      @{lemma "a \<notin> (S - T) \<longleftrightarrow> a \<notin> S \<or> a \<in> T" by simp}, 

      @{lemma "(lhs = (a \<notin> {})) \<longleftrightarrow> lhs" by simp}, 

      @{thm fresh_def[symmetric]}]

  val qfresh_constrs = qsupp_constrs

    |> map (fn thm => thm RS transform_thm) 

    |> map (simplify (HOL_basic_ss addsimps transform_thms))

  (* proving that the qbn result is finite *)

  val qbn_finite_thms = prove_bns_finite qtys qbns qinduct qbn_defs lthyC

  (* proving that perm_bns preserve alpha *)

  val qperm_bn_alpha_thms = 

    prove_perm_bn_alpha_thms qtys qperm_bns qalpha_bns qinduct qperm_bn_simps qeq_iffs' 

      qalpha_refl_thms lthyC

  (* proving the relationship of bn and permute_bn *)

  val qpermute_bn_thms = 

    prove_permute_bn_thms qtys qbns qperm_bns qinduct qperm_bn_simps qbn_defs qfv_qbn_eqvts lthyC

  val _ = trace_msg (K "Proving strong exhaust lemmas...")

  val qstrong_exhaust_thms = prove_strong_exhausts lthyC qexhausts bclauses qbn_finite_thms qeq_iffs'

    qfv_qbn_eqvts qpermute_bn_thms qperm_bn_alpha_thms

  val _ = trace_msg (K "Proving strong induct lemmas...")

  val qstrong_induct_thms =  prove_strong_induct lthyC qinduct qstrong_exhaust_thms qsize_simps bclauses

  (* noting the theorems *)  

  (* generating the prefix for the theorem names *)

  val thms_name = 

    the_default (Binding.name (space_implode "_" qty_names)) opt_thms_name 

  fun thms_suffix s = Binding.qualified true s thms_name 

  val case_names_attr = Attrib.internal (K (Rule_Cases.case_names cnstr_names))

  val infos = mk_infos qty_full_names qeq_iffs' qdistincts qstrong_exhaust_thms qstrong_induct_thms

  val (_, lthy9') = lthyC

     |> Local_Theory.declaration false (K (fold register_info infos))

     |> Local_Theory.note ((thms_suffix "distinct", [induct_attr, simp_attr]), qdistincts) 

     ||>> Local_Theory.note ((thms_suffix "eq_iff", [induct_attr, simp_attr]), qeq_iffs')

     ||>> Local_Theory.note ((thms_suffix "fv_defs", []), qfv_defs) 

     ||>> Local_Theory.note ((thms_suffix "bn_defs", []), qbn_defs) 

     ||>> Local_Theory.note ((thms_suffix "perm_simps", [eqvt_attr, simp_attr]), qperm_simps) 

     ||>> Local_Theory.note ((thms_suffix "fv_bn_eqvt", []), qfv_qbn_eqvts) 

     ||>> Local_Theory.note ((thms_suffix "size", [simp_attr]), qsize_simps)

     ||>> Local_Theory.note ((thms_suffix "size_eqvt", []), qsize_eqvt)

     ||>> Local_Theory.note ((thms_suffix "induct", [case_names_attr]), [qinduct]) 

     ||>> Local_Theory.note ((thms_suffix "inducts", [case_names_attr]), qinducts)

     ||>> Local_Theory.note ((thms_suffix "exhaust", [case_names_attr]), qexhausts)

     ||>> Local_Theory.note ((thms_suffix "strong_exhaust", [case_names_attr]), qstrong_exhaust_thms)

     ||>> Local_Theory.note ((thms_suffix "strong_induct", [case_names_attr]), qstrong_induct_thms)

     ||>> Local_Theory.note ((thms_suffix "supports", []), qsupports_thms)

     ||>> Local_Theory.note ((thms_suffix "fsupp", []), qfsupp_thms)

     ||>> Local_Theory.note ((thms_suffix "supp", []), qsupp_constrs)