(*  Title:      HOL/HOLCF/Tools/Domain/domain_constructors.ML

     Author:     Brian Huffman

 Defines constructor functions for a given domain isomorphism

 and proves related theorems.

 *)

 signature DOMAIN_CONSTRUCTORS =

 sig

   type constr_info =

     {

       iso_info : Domain_Take_Proofs.iso_info,

       con_specs : (term * (bool * typ) list) list,

       con_betas : thm list,

       nchotomy : thm,

       exhaust : thm,

       compacts : thm list,

       con_rews : thm list,

       inverts : thm list,

       injects : thm list,

       dist_les : thm list,

       dist_eqs : thm list,

       cases : thm list,

       sel_rews : thm list,

       dis_rews : thm list,

       match_rews : thm list

     }

   val add_domain_constructors :

       binding

       -> (binding * (bool * binding option * typ) list * mixfix) list

       -> Domain_Take_Proofs.iso_info

       -> theory

       -> constr_info * theory

 end

 structure Domain_Constructors : DOMAIN_CONSTRUCTORS =

 struct

 open HOLCF_Library

 infixr 6 ->>

 infix -->>

 infix 9 `

 type constr_info =

   {

     iso_info : Domain_Take_Proofs.iso_info,

     con_specs : (term * (bool * typ) list) list,

     con_betas : thm list,

     nchotomy : thm,

     exhaust : thm,

     compacts : thm list,

     con_rews : thm list,

     inverts : thm list,

     injects : thm list,

     dist_les : thm list,

     dist_eqs : thm list,

     cases : thm list,

     sel_rews : thm list,

     dis_rews : thm list,

     match_rews : thm list

   }

 (************************** miscellaneous functions ***************************)

 val simple_ss = HOL_basic_ss addsimps simp_thms

 val beta_rules =

   @{thms beta_cfun cont_id cont_const cont2cont_APP cont2cont_LAM'} @

   @{thms cont2cont_fst cont2cont_snd cont2cont_Pair}

 val beta_ss = HOL_basic_ss addsimps (simp_thms @ beta_rules)

 fun define_consts

     (specs : (binding * term * mixfix) list)

     (thy : theory)

     : (term list * thm list) * theory =

   let

     fun mk_decl (b, t, mx) = (b, fastype_of t, mx)

     val decls = map mk_decl specs

     val thy = Cont_Consts.add_consts decls thy

     fun mk_const (b, T, mx) = Const (Sign.full_name thy b, T)

     val consts = map mk_const decls

     fun mk_def c (b, t, mx) =

       (Binding.suffix_name "_def" b, Logic.mk_equals (c, t))

     val defs = map2 mk_def consts specs

     val (def_thms, thy) =

       Global_Theory.add_defs false (map Thm.no_attributes defs) thy

   in

     ((consts, def_thms), thy)

   end

 fun prove

     (thy : theory)

     (defs : thm list)

     (goal : term)

     (tacs : {prems: thm list, context: Proof.context} -> tactic list)

     : thm =

   let

     fun tac {prems, context} =

       rewrite_goals_tac defs THEN

       EVERY (tacs {prems = map (rewrite_rule defs) prems, context = context})

   in

     Goal.prove_global thy [] [] goal tac

   end

 fun get_vars_avoiding

     (taken : string list)

     (args : (bool * typ) list)

     : (term list * term list) =

   let

     val Ts = map snd args

     val ns = Name.variant_list taken (Datatype_Prop.make_tnames Ts)

     val vs = map Free (ns ~~ Ts)

     val nonlazy = map snd (filter_out (fst o fst) (args ~~ vs))

   in

     (vs, nonlazy)

   end

 fun get_vars args = get_vars_avoiding [] args

 (************** generating beta reduction rules from definitions **************)

 local

   fun arglist (Const _ $ Abs (s, T, t)) =

       let

         val arg = Free (s, T)

         val (args, body) = arglist (subst_bound (arg, t))

       in (arg :: args, body) end

     | arglist t = ([], t)

 in

   fun beta_of_def thy def_thm =

       let

         val (con, lam) = Logic.dest_equals (concl_of def_thm)

         val (args, rhs) = arglist lam

         val lhs = list_ccomb (con, args)

         val goal = mk_equals (lhs, rhs)

         val cs = ContProc.cont_thms lam

         val betas = map (fn c => mk_meta_eq (c RS @{thm beta_cfun})) cs

       in

         prove thy (def_thm::betas) goal (K [rtac reflexive_thm 1])

       end

 end

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

 (************* definitions and theorems for constructor functions *************)

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

 fun add_constructors

     (spec : (binding * (bool * typ) list * mixfix) list)

     (abs_const : term)

     (iso_locale : thm)

     (thy : theory)

     =

   let

     (* get theorems about rep and abs *)

     val abs_strict = iso_locale RS @{thm iso.abs_strict}

     (* get types of type isomorphism *)

     val (rhsT, lhsT) = dest_cfunT (fastype_of abs_const)

     fun vars_of args =

       let

         val Ts = map snd args

         val ns = Datatype_Prop.make_tnames Ts

       in

         map Free (ns ~~ Ts)

       end

     (* define constructor functions *)

     val ((con_consts, con_defs), thy) =

       let

         fun one_arg (lazy, T) var = if lazy then mk_up var else var

         fun one_con (_,args,_) = mk_stuple (map2 one_arg args (vars_of args))

         fun mk_abs t = abs_const ` t

         val rhss = map mk_abs (mk_sinjects (map one_con spec))

         fun mk_def (bind, args, mx) rhs =

           (bind, big_lambdas (vars_of args) rhs, mx)

       in

         define_consts (map2 mk_def spec rhss) thy

       end

     (* prove beta reduction rules for constructors *)

     val con_betas = map (beta_of_def thy) con_defs

     (* replace bindings with terms in constructor spec *)

     val spec' : (term * (bool * typ) list) list =

       let fun one_con con (b, args, mx) = (con, args)

       in map2 one_con con_consts spec end

     (* prove exhaustiveness of constructors *)

     local

       fun arg2typ n (true,  T) = (n+1, mk_upT (TVar (("'a", n), @{sort cpo})))

         | arg2typ n (false, T) = (n+1, TVar (("'a", n), @{sort pcpo}))

       fun args2typ n [] = (n, oneT)

         | args2typ n [arg] = arg2typ n arg

         | args2typ n (arg::args) =

           let

             val (n1, t1) = arg2typ n arg

             val (n2, t2) = args2typ n1 args

           in (n2, mk_sprodT (t1, t2)) end

       fun cons2typ n [] = (n, oneT)

         | cons2typ n [con] = args2typ n (snd con)

         | cons2typ n (con::cons) =

           let

             val (n1, t1) = args2typ n (snd con)

             val (n2, t2) = cons2typ n1 cons

           in (n2, mk_ssumT (t1, t2)) end

       val ct = ctyp_of thy (snd (cons2typ 1 spec'))

       val thm1 = instantiate' [SOME ct] [] @{thm exh_start}

       val thm2 = rewrite_rule (map mk_meta_eq @{thms ex_bottom_iffs}) thm1

       val thm3 = rewrite_rule [mk_meta_eq @{thm conj_assoc}] thm2

       val y = Free ("y", lhsT)

       fun one_con (con, args) =

         let

           val (vs, nonlazy) = get_vars_avoiding ["y"] args

           val eqn = mk_eq (y, list_ccomb (con, vs))

           val conj = foldr1 mk_conj (eqn :: map mk_defined nonlazy)

         in Library.foldr mk_ex (vs, conj) end

       val goal = mk_trp (foldr1 mk_disj (mk_undef y :: map one_con spec'))

       (* first rules replace "y = bottom \/ P" with "rep$y = bottom \/ P" *)

       val tacs = [

           rtac (iso_locale RS @{thm iso.casedist_rule}) 1,

           rewrite_goals_tac [mk_meta_eq (iso_locale RS @{thm iso.iso_swap})],

           rtac thm3 1]

     in

       val nchotomy = prove thy con_betas goal (K tacs)

       val exhaust =

           (nchotomy RS @{thm exh_casedist0})

           |> rewrite_rule @{thms exh_casedists}

           |> Drule.zero_var_indexes

     end

     (* prove compactness rules for constructors *)

     val compacts =

       let

         val rules = @{thms compact_sinl compact_sinr compact_spair

                            compact_up compact_ONE}

         val tacs =

           [rtac (iso_locale RS @{thm iso.compact_abs}) 1,

            REPEAT (resolve_tac rules 1 ORELSE atac 1)]

         fun con_compact (con, args) =

           let

             val vs = vars_of args

             val con_app = list_ccomb (con, vs)

             val concl = mk_trp (mk_compact con_app)

             val assms = map (mk_trp o mk_compact) vs

             val goal = Logic.list_implies (assms, concl)

           in

             prove thy con_betas goal (K tacs)

           end

       in

         map con_compact spec'

       end

     (* prove strictness rules for constructors *)

     local

       fun con_strict (con, args) = 

         let

           val rules = abs_strict :: @{thms con_strict_rules}

           val (vs, nonlazy) = get_vars args

           fun one_strict v' =

             let

               val bottom = mk_bottom (fastype_of v')

               val vs' = map (fn v => if v = v' then bottom else v) vs

               val goal = mk_trp (mk_undef (list_ccomb (con, vs')))

               val tacs = [simp_tac (HOL_basic_ss addsimps rules) 1]

             in prove thy con_betas goal (K tacs) end

         in map one_strict nonlazy end

       fun con_defin (con, args) =

         let

           fun iff_disj (t, []) = HOLogic.mk_not t

             | iff_disj (t, ts) = mk_eq (t, foldr1 HOLogic.mk_disj ts)

           val (vs, nonlazy) = get_vars args

           val lhs = mk_undef (list_ccomb (con, vs))

           val rhss = map mk_undef nonlazy

           val goal = mk_trp (iff_disj (lhs, rhss))

           val rule1 = iso_locale RS @{thm iso.abs_bottom_iff}

           val rules = rule1 :: @{thms con_bottom_iff_rules}

           val tacs = [simp_tac (HOL_ss addsimps rules) 1]

         in prove thy con_betas goal (K tacs) end

     in

       val con_stricts = maps con_strict spec'

       val con_defins = map con_defin spec'

       val con_rews = con_stricts @ con_defins

     end

     (* prove injectiveness of constructors *)

     local

       fun pgterm rel (con, args) =

         let

           fun prime (Free (n, T)) = Free (n^"'", T)

             | prime t             = t

           val (xs, nonlazy) = get_vars args

           val ys = map prime xs

           val lhs = rel (list_ccomb (con, xs), list_ccomb (con, ys))

           val rhs = foldr1 mk_conj (ListPair.map rel (xs, ys))

           val concl = mk_trp (mk_eq (lhs, rhs))

           val zs = case args of [_] => [] | _ => nonlazy

           val assms = map (mk_trp o mk_defined) zs

           val goal = Logic.list_implies (assms, concl)

         in prove thy con_betas goal end

       val cons' = filter (fn (_, args) => not (null args)) spec'

     in

       val inverts =

         let

           val abs_below = iso_locale RS @{thm iso.abs_below}

           val rules1 = abs_below :: @{thms sinl_below sinr_below spair_below up_below}

           val rules2 = @{thms up_defined spair_defined ONE_defined}

           val rules = rules1 @ rules2

           val tacs = [asm_simp_tac (simple_ss addsimps rules) 1]

         in map (fn c => pgterm mk_below c (K tacs)) cons' end

       val injects =

         let

           val abs_eq = iso_locale RS @{thm iso.abs_eq}

           val rules1 = abs_eq :: @{thms sinl_eq sinr_eq spair_eq up_eq}

           val rules2 = @{thms up_defined spair_defined ONE_defined}

           val rules = rules1 @ rules2

           val tacs = [asm_simp_tac (simple_ss addsimps rules) 1]

         in map (fn c => pgterm mk_eq c (K tacs)) cons' end

     end

     (* prove distinctness of constructors *)

     local

       fun map_dist (f : 'a -> 'a -> 'b) (xs : 'a list) : 'b list =

         flat (map_index (fn (i, x) => map (f x) (nth_drop i xs)) xs)

       fun prime (Free (n, T)) = Free (n^"'", T)

         | prime t             = t

       fun iff_disj (t, []) = mk_not t

         | iff_disj (t, ts) = mk_eq (t, foldr1 mk_disj ts)

       fun iff_disj2 (t, [], us) = mk_not t

         | iff_disj2 (t, ts, []) = mk_not t

         | iff_disj2 (t, ts, us) =

           mk_eq (t, mk_conj (foldr1 mk_disj ts, foldr1 mk_disj us))

       fun dist_le (con1, args1) (con2, args2) =

         let

           val (vs1, zs1) = get_vars args1

           val (vs2, zs2) = get_vars args2 |> pairself (map prime)

           val lhs = mk_below (list_ccomb (con1, vs1), list_ccomb (con2, vs2))

           val rhss = map mk_undef zs1

           val goal = mk_trp (iff_disj (lhs, rhss))

           val rule1 = iso_locale RS @{thm iso.abs_below}

           val rules = rule1 :: @{thms con_below_iff_rules}

           val tacs = [simp_tac (HOL_ss addsimps rules) 1]

         in prove thy con_betas goal (K tacs) end

       fun dist_eq (con1, args1) (con2, args2) =

         let

           val (vs1, zs1) = get_vars args1

           val (vs2, zs2) = get_vars args2 |> pairself (map prime)

           val lhs = mk_eq (list_ccomb (con1, vs1), list_ccomb (con2, vs2))

           val rhss1 = map mk_undef zs1

           val rhss2 = map mk_undef zs2

           val goal = mk_trp (iff_disj2 (lhs, rhss1, rhss2))

           val rule1 = iso_locale RS @{thm iso.abs_eq}

           val rules = rule1 :: @{thms con_eq_iff_rules}

           val tacs = [simp_tac (HOL_ss addsimps rules) 1]

         in prove thy con_betas goal (K tacs) end

     in

       val dist_les = map_dist dist_le spec'

       val dist_eqs = map_dist dist_eq spec'

     end

     val result =

       {

         con_consts = con_consts,

         con_betas = con_betas,

         nchotomy = nchotomy,

         exhaust = exhaust,

         compacts = compacts,

         con_rews = con_rews,

         inverts = inverts,

         injects = injects,

         dist_les = dist_les,

         dist_eqs = dist_eqs

       }

   in

     (result, thy)

   end

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

 (**************** definition and theorems for case combinator *****************)

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

 fun add_case_combinator

     (spec : (term * (bool * typ) list) list)

     (lhsT : typ)

     (dbind : binding)

     (con_betas : thm list)

     (exhaust : thm)

     (iso_locale : thm)

     (rep_const : term)

     (thy : theory)

     : ((typ -> term) * thm list) * theory =

   let

     (* prove rep/abs rules *)

     val rep_strict = iso_locale RS @{thm iso.rep_strict}

     val abs_inverse = iso_locale RS @{thm iso.abs_iso}

     (* calculate function arguments of case combinator *)

     val tns = map fst (Term.add_tfreesT lhsT [])

     val resultT = TFree (singleton (Name.variant_list tns) "'t", @{sort pcpo})

     fun fTs T = map (fn (_, args) => map snd args -->> T) spec

     val fns = Datatype_Prop.indexify_names (map (K "f") spec)

     val fs = map Free (fns ~~ fTs resultT)

     fun caseT T = fTs T -->> (lhsT ->> T)

     (* definition of case combinator *)

     local

       val case_bind = Binding.suffix_name "_case" dbind

       fun lambda_arg (lazy, v) t =

           (if lazy then mk_fup else I) (big_lambda v t)

       fun lambda_args []      t = mk_one_case t

         | lambda_args (x::[]) t = lambda_arg x t

         | lambda_args (x::xs) t = mk_ssplit (lambda_arg x (lambda_args xs t))

       fun one_con f (_, args) =

         let

           val Ts = map snd args

           val ns = Name.variant_list fns (Datatype_Prop.make_tnames Ts)

           val vs = map Free (ns ~~ Ts)

         in

           lambda_args (map fst args ~~ vs) (list_ccomb (f, vs))

         end

       fun mk_sscases [t] = mk_strictify t

         | mk_sscases ts = foldr1 mk_sscase ts

       val body = mk_sscases (map2 one_con fs spec)

       val rhs = big_lambdas fs (mk_cfcomp (body, rep_const))

       val ((case_consts, case_defs), thy) =

           define_consts [(case_bind, rhs, NoSyn)] thy

       val case_name = Sign.full_name thy case_bind

     in

       val case_def = hd case_defs

       fun case_const T = Const (case_name, caseT T)

       val case_app = list_ccomb (case_const resultT, fs)

       val thy = thy

     end

     (* define syntax for case combinator *)

     (* TODO: re-implement case syntax using a parse translation *)

     local

       fun syntax c = Lexicon.mark_const (fst (dest_Const c))

       fun xconst c = Long_Name.base_name (fst (dest_Const c))

       fun c_ast authentic con = Ast.Constant (if authentic then syntax con else xconst con)

       fun showint n = string_of_int (n+1)

       fun expvar n = Ast.Variable ("e" ^ showint n)

       fun argvar n (m, _) = Ast.Variable ("a" ^ showint n ^ "_" ^ showint m)

       fun argvars n args = map_index (argvar n) args

       fun app s (l, r) = Ast.mk_appl (Ast.Constant s) [l, r]

       val cabs = app "_cabs"

       val capp = app @{const_syntax Rep_cfun}

       val capps = Library.foldl capp

       fun con1 authentic n (con,args) =

           Library.foldl capp (c_ast authentic con, argvars n args)

       fun case1 authentic (n, c) =

           app "_case1" (Ast.strip_positions (con1 authentic n c), expvar n)

       fun arg1 (n, (con,args)) = List.foldr cabs (expvar n) (argvars n args)

       fun when1 n (m, c) = if n = m then arg1 (n, c) else Ast.Constant @{const_syntax bottom}

       val case_constant = Ast.Constant (syntax (case_const dummyT))

       fun case_trans authentic =

         (if authentic then Syntax.Parse_Print_Rule else Syntax.Parse_Rule)

           (app "_case_syntax"

             (Ast.Variable "x",

              foldr1 (app "_case2") (map_index (case1 authentic) spec)),

            capp (capps (case_constant, map_index arg1 spec), Ast.Variable "x"))

       fun one_abscon_trans authentic (n, c) =

         (if authentic then Syntax.Parse_Print_Rule else Syntax.Parse_Rule)

           (cabs (con1 authentic n c, expvar n),

            capps (case_constant, map_index (when1 n) spec))

       fun abscon_trans authentic =

           map_index (one_abscon_trans authentic) spec

       val trans_rules : Ast.ast Syntax.trrule list =

           case_trans false :: case_trans true ::

           abscon_trans false @ abscon_trans true

     in

       val thy = Sign.add_trrules trans_rules thy

     end

     (* prove beta reduction rule for case combinator *)

     val case_beta = beta_of_def thy case_def

     (* prove strictness of case combinator *)

     val case_strict =

       let

         val defs = case_beta :: map mk_meta_eq [rep_strict, @{thm cfcomp2}]

         val goal = mk_trp (mk_strict case_app)

         val rules = @{thms sscase1 ssplit1 strictify1 one_case1}

         val tacs = [resolve_tac rules 1]

       in prove thy defs goal (K tacs) end

     (* prove rewrites for case combinator *)

     local

       fun one_case (con, args) f =

         let

           val (vs, nonlazy) = get_vars args

           val assms = map (mk_trp o mk_defined) nonlazy

           val lhs = case_app ` list_ccomb (con, vs)

           val rhs = list_ccomb (f, vs)

           val concl = mk_trp (mk_eq (lhs, rhs))

           val goal = Logic.list_implies (assms, concl)

           val defs = case_beta :: con_betas

           val rules1 = @{thms strictify2 sscase2 sscase3 ssplit2 fup2 ID1}

           val rules2 = @{thms con_bottom_iff_rules}

           val rules3 = @{thms cfcomp2 one_case2}

           val rules = abs_inverse :: rules1 @ rules2 @ rules3

           val tacs = [asm_simp_tac (beta_ss addsimps rules) 1]

         in prove thy defs goal (K tacs) end

     in

       val case_apps = map2 one_case spec fs

     end

   in

     ((case_const, case_strict :: case_apps), thy)

   end

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

 (************** definitions and theorems for selector functions ***************)

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

 fun add_selectors

     (spec : (term * (bool * binding option * typ) list) list)

     (rep_const : term)

     (abs_inv : thm)

     (rep_strict : thm)

     (rep_bottom_iff : thm)

     (con_betas : thm list)

     (thy : theory)

     : thm list * theory =

   let

     (* define selector functions *)

     val ((sel_consts, sel_defs), thy) =

       let

         fun rangeT s = snd (dest_cfunT (fastype_of s))

         fun mk_outl s = mk_cfcomp (from_sinl (dest_ssumT (rangeT s)), s)

         fun mk_outr s = mk_cfcomp (from_sinr (dest_ssumT (rangeT s)), s)

         fun mk_sfst s = mk_cfcomp (sfst_const (dest_sprodT (rangeT s)), s)

         fun mk_ssnd s = mk_cfcomp (ssnd_const (dest_sprodT (rangeT s)), s)

         fun mk_down s = mk_cfcomp (from_up (dest_upT (rangeT s)), s)

         fun sels_of_arg s (lazy, NONE,   T) = []

           | sels_of_arg s (lazy, SOME b, T) =

             [(b, if lazy then mk_down s else s, NoSyn)]

         fun sels_of_args s [] = []

           | sels_of_args s (v :: []) = sels_of_arg s v

           | sels_of_args s (v :: vs) =

             sels_of_arg (mk_sfst s) v @ sels_of_args (mk_ssnd s) vs

         fun sels_of_cons s [] = []

           | sels_of_cons s ((con, args) :: []) = sels_of_args s args

           | sels_of_cons s ((con, args) :: cs) =

             sels_of_args (mk_outl s) args @ sels_of_cons (mk_outr s) cs

         val sel_eqns : (binding * term * mixfix) list =

             sels_of_cons rep_const spec

       in

         define_consts sel_eqns thy

       end

     (* replace bindings with terms in constructor spec *)

     val spec2 : (term * (bool * term option * typ) list) list =

       let

         fun prep_arg (lazy, NONE, T) sels = ((lazy, NONE, T), sels)

           | prep_arg (lazy, SOME _, T) sels =

             ((lazy, SOME (hd sels), T), tl sels)

         fun prep_con (con, args) sels =

             apfst (pair con) (fold_map prep_arg args sels)

       in

         fst (fold_map prep_con spec sel_consts)

       end

     (* prove selector strictness rules *)

     val sel_stricts : thm list =

       let

         val rules = rep_strict :: @{thms sel_strict_rules}

         val tacs = [simp_tac (HOL_basic_ss addsimps rules) 1]

         fun sel_strict sel =

           let

             val goal = mk_trp (mk_strict sel)

           in

             prove thy sel_defs goal (K tacs)

           end

       in

         map sel_strict sel_consts

       end

     (* prove selector application rules *)

     val sel_apps : thm list =

       let

         val defs = con_betas @ sel_defs

         val rules = abs_inv :: @{thms sel_app_rules}

         val tacs = [asm_simp_tac (simple_ss addsimps rules) 1]

         fun sel_apps_of (i, (con, args: (bool * term option * typ) list)) =

           let

             val Ts : typ list = map #3 args

             val ns : string list = Datatype_Prop.make_tnames Ts

             val vs : term list = map Free (ns ~~ Ts)

             val con_app : term = list_ccomb (con, vs)

             val vs' : (bool * term) list = map #1 args ~~ vs

             fun one_same (n, sel, T) =

               let

                 val xs = map snd (filter_out fst (nth_drop n vs'))

                 val assms = map (mk_trp o mk_defined) xs

                 val concl = mk_trp (mk_eq (sel ` con_app, nth vs n))

                 val goal = Logic.list_implies (assms, concl)

               in

                 prove thy defs goal (K tacs)

               end

             fun one_diff (n, sel, T) =

               let

                 val goal = mk_trp (mk_eq (sel ` con_app, mk_bottom T))

               in

                 prove thy defs goal (K tacs)

               end

             fun one_con (j, (_, args')) : thm list =

               let

                 fun prep (i, (lazy, NONE, T)) = NONE

                   | prep (i, (lazy, SOME sel, T)) = SOME (i, sel, T)

                 val sels : (int * term * typ) list =

                   map_filter prep (map_index I args')

               in

                 if i = j

                 then map one_same sels

                 else map one_diff sels

               end

           in

             flat (map_index one_con spec2)

           end

       in

         flat (map_index sel_apps_of spec2)

       end

   (* prove selector definedness rules *)

     val sel_defins : thm list =

       let

         val rules = rep_bottom_iff :: @{thms sel_bottom_iff_rules}

         val tacs = [simp_tac (HOL_basic_ss addsimps rules) 1]

         fun sel_defin sel =

           let

             val (T, U) = dest_cfunT (fastype_of sel)

             val x = Free ("x", T)

             val lhs = mk_eq (sel ` x, mk_bottom U)

             val rhs = mk_eq (x, mk_bottom T)

             val goal = mk_trp (mk_eq (lhs, rhs))

           in

             prove thy sel_defs goal (K tacs)

           end

         fun one_arg (false, SOME sel, T) = SOME (sel_defin sel)

           | one_arg _                    = NONE

       in

         case spec2 of

           [(con, args)] => map_filter one_arg args

         | _             => []

       end

   in

     (sel_stricts @ sel_defins @ sel_apps, thy)

   end

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

 (************ definitions and theorems for discriminator functions ************)

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

 fun add_discriminators

     (bindings : binding list)

     (spec : (term * (bool * typ) list) list)

     (lhsT : typ)

     (exhaust : thm)

     (case_const : typ -> term)

     (case_rews : thm list)

     (thy : theory) =

   let

     fun vars_of args =

       let

         val Ts = map snd args

         val ns = Datatype_Prop.make_tnames Ts

       in

         map Free (ns ~~ Ts)

       end

     (* define discriminator functions *)

     local

       fun dis_fun i (j, (con, args)) =

         let

           val (vs, nonlazy) = get_vars args

           val tr = if i = j then @{term TT} else @{term FF}

         in

           big_lambdas vs tr

         end

       fun dis_eqn (i, bind) : binding * term * mixfix =

         let

           val dis_bind = Binding.prefix_name "is_" bind

           val rhs = list_ccomb (case_const trT, map_index (dis_fun i) spec)

         in

           (dis_bind, rhs, NoSyn)

         end

     in

       val ((dis_consts, dis_defs), thy) =

           define_consts (map_index dis_eqn bindings) thy

     end

     (* prove discriminator strictness rules *)

     local

       fun dis_strict dis =

         let val goal = mk_trp (mk_strict dis)

         in prove thy dis_defs goal (K [rtac (hd case_rews) 1]) end

     in

       val dis_stricts = map dis_strict dis_consts

     end

     (* prove discriminator/constructor rules *)

     local

       fun dis_app (i, dis) (j, (con, args)) =

         let

           val (vs, nonlazy) = get_vars args

           val lhs = dis ` list_ccomb (con, vs)

           val rhs = if i = j then @{term TT} else @{term FF}

           val assms = map (mk_trp o mk_defined) nonlazy

           val concl = mk_trp (mk_eq (lhs, rhs))

           val goal = Logic.list_implies (assms, concl)

           val tacs = [asm_simp_tac (beta_ss addsimps case_rews) 1]

         in prove thy dis_defs goal (K tacs) end

       fun one_dis (i, dis) =

           map_index (dis_app (i, dis)) spec

     in

       val dis_apps = flat (map_index one_dis dis_consts)

     end

     (* prove discriminator definedness rules *)

     local

       fun dis_defin dis =

         let

           val x = Free ("x", lhsT)

           val simps = dis_apps @ @{thms dist_eq_tr}

           val tacs =

             [rtac @{thm iffI} 1,

              asm_simp_tac (HOL_basic_ss addsimps dis_stricts) 2,

              rtac exhaust 1, atac 1,

              DETERM_UNTIL_SOLVED (CHANGED

                (asm_full_simp_tac (simple_ss addsimps simps) 1))]

           val goal = mk_trp (mk_eq (mk_undef (dis ` x), mk_undef x))

         in prove thy [] goal (K tacs) end

     in

       val dis_defins = map dis_defin dis_consts

     end

   in

     (dis_stricts @ dis_defins @ dis_apps, thy)

   end

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

 (*************** definitions and theorems for match combinators ***************)

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

 fun add_match_combinators

     (bindings : binding list)

     (spec : (term * (bool * typ) list) list)

     (lhsT : typ)

     (exhaust : thm)

     (case_const : typ -> term)

     (case_rews : thm list)

     (thy : theory) =

   let

     (* get a fresh type variable for the result type *)

     val resultT : typ =

       let

         val ts : string list = map fst (Term.add_tfreesT lhsT [])

         val t : string = singleton (Name.variant_list ts) "'t"

       in TFree (t, @{sort pcpo}) end

     (* define match combinators *)

     local

       val x = Free ("x", lhsT)

       fun k args = Free ("k", map snd args -->> mk_matchT resultT)

       val fail = mk_fail resultT

       fun mat_fun i (j, (con, args)) =

         let

           val (vs, nonlazy) = get_vars_avoiding ["x","k"] args

         in

           if i = j then k args else big_lambdas vs fail

         end

       fun mat_eqn (i, (bind, (con, args))) : binding * term * mixfix =

         let

           val mat_bind = Binding.prefix_name "match_" bind

           val funs = map_index (mat_fun i) spec

           val body = list_ccomb (case_const (mk_matchT resultT), funs)

           val rhs = big_lambda x (big_lambda (k args) (body ` x))

         in

           (mat_bind, rhs, NoSyn)

         end

     in

       val ((match_consts, match_defs), thy) =

           define_consts (map_index mat_eqn (bindings ~~ spec)) thy

     end

     (* register match combinators with fixrec package *)

     local

       val con_names = map (fst o dest_Const o fst) spec

       val mat_names = map (fst o dest_Const) match_consts

     in

       val thy = Fixrec.add_matchers (con_names ~~ mat_names) thy

     end

     (* prove strictness of match combinators *)

     local

       fun match_strict mat =

         let

           val (T, (U, V)) = apsnd dest_cfunT (dest_cfunT (fastype_of mat))

           val k = Free ("k", U)

           val goal = mk_trp (mk_eq (mat ` mk_bottom T ` k, mk_bottom V))

           val tacs = [asm_simp_tac (beta_ss addsimps case_rews) 1]

         in prove thy match_defs goal (K tacs) end

     in

       val match_stricts = map match_strict match_consts

     end

     (* prove match/constructor rules *)

     local

       val fail = mk_fail resultT

       fun match_app (i, mat) (j, (con, args)) =

         let

           val (vs, nonlazy) = get_vars_avoiding ["k"] args

           val (_, (kT, _)) = apsnd dest_cfunT (dest_cfunT (fastype_of mat))

           val k = Free ("k", kT)

           val lhs = mat ` list_ccomb (con, vs) ` k

           val rhs = if i = j then list_ccomb (k, vs) else fail

           val assms = map (mk_trp o mk_defined) nonlazy

           val concl = mk_trp (mk_eq (lhs, rhs))

           val goal = Logic.list_implies (assms, concl)

           val tacs = [asm_simp_tac (beta_ss addsimps case_rews) 1]

         in prove thy match_defs goal (K tacs) end

       fun one_match (i, mat) =

           map_index (match_app (i, mat)) spec

     in

       val match_apps = flat (map_index one_match match_consts)

     end

   in

     (match_stricts @ match_apps, thy)

   end

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

 (******************************* main function ********************************)

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

 fun add_domain_constructors

     (dbind : binding)

     (spec : (binding * (bool * binding option * typ) list * mixfix) list)

     (iso_info : Domain_Take_Proofs.iso_info)

     (thy : theory) =

   let

     val dname = Binding.name_of dbind

     val _ = writeln ("Proving isomorphism properties of domain "^dname^" ...")

     val bindings = map #1 spec

     (* retrieve facts about rep/abs *)

     val lhsT = #absT iso_info

     val {rep_const, abs_const, ...} = iso_info

     val abs_iso_thm = #abs_inverse iso_info

     val rep_iso_thm = #rep_inverse iso_info

     val iso_locale = @{thm iso.intro} OF [abs_iso_thm, rep_iso_thm]

     val rep_strict = iso_locale RS @{thm iso.rep_strict}

     val abs_strict = iso_locale RS @{thm iso.abs_strict}

     val rep_bottom_iff = iso_locale RS @{thm iso.rep_bottom_iff}

     val abs_bottom_iff = iso_locale RS @{thm iso.abs_bottom_iff}

     val iso_rews = [abs_iso_thm, rep_iso_thm, abs_strict, rep_strict]

     (* qualify constants and theorems with domain name *)

     val thy = Sign.add_path dname thy

     (* define constructor functions *)

     val (con_result, thy) =

       let

         fun prep_arg (lazy, sel, T) = (lazy, T)

         fun prep_con (b, args, mx) = (b, map prep_arg args, mx)

         val con_spec = map prep_con spec

       in

         add_constructors con_spec abs_const iso_locale thy

       end

     val {con_consts, con_betas, nchotomy, exhaust, compacts, con_rews,

           inverts, injects, dist_les, dist_eqs} = con_result

     (* prepare constructor spec *)

     val con_specs : (term * (bool * typ) list) list =

       let

         fun prep_arg (lazy, sel, T) = (lazy, T)

         fun prep_con c (b, args, mx) = (c, map prep_arg args)

       in

         map2 prep_con con_consts spec

       end

     (* define case combinator *)

     val ((case_const : typ -> term, cases : thm list), thy) =

         add_case_combinator con_specs lhsT dbind

           con_betas exhaust iso_locale rep_const thy

     (* define and prove theorems for selector functions *)

     val (sel_thms : thm list, thy : theory) =

       let

         val sel_spec : (term * (bool * binding option * typ) list) list =

           map2 (fn con => fn (b, args, mx) => (con, args)) con_consts spec

       in

         add_selectors sel_spec rep_const

           abs_iso_thm rep_strict rep_bottom_iff con_betas thy

       end

     (* define and prove theorems for discriminator functions *)

     val (dis_thms : thm list, thy : theory) =

         add_discriminators bindings con_specs lhsT

           exhaust case_const cases thy

     (* define and prove theorems for match combinators *)

     val (match_thms : thm list, thy : theory) =

         add_match_combinators bindings con_specs lhsT

           exhaust case_const cases thy

     (* restore original signature path *)

     val thy = Sign.parent_path thy

     (* bind theorem names in global theory *)

     val (_, thy) =

       let

         fun qualified name = Binding.qualified true name dbind

         val names = "bottom" :: map (fn (b,_,_) => Binding.name_of b) spec

         val dname = fst (dest_Type lhsT)

         val simp = Simplifier.simp_add

         val case_names = Rule_Cases.case_names names

         val cases_type = Induct.cases_type dname

       in

         Global_Theory.add_thmss [

           ((qualified "iso_rews"  , iso_rews    ), [simp]),

           ((qualified "nchotomy"  , [nchotomy]  ), []),

           ((qualified "exhaust"   , [exhaust]   ), [case_names, cases_type]),

           ((qualified "case_rews" , cases       ), [simp]),

           ((qualified "compacts"  , compacts    ), [simp]),

           ((qualified "con_rews"  , con_rews    ), [simp]),

           ((qualified "sel_rews"  , sel_thms    ), [simp]),

           ((qualified "dis_rews"  , dis_thms    ), [simp]),

           ((qualified "dist_les"  , dist_les    ), [simp]),

           ((qualified "dist_eqs"  , dist_eqs    ), [simp]),

           ((qualified "inverts"   , inverts     ), [simp]),

           ((qualified "injects"   , injects     ), [simp]),

           ((qualified "match_rews", match_thms  ), [simp])] thy

       end

     val result =

       {

         iso_info = iso_info,

         con_specs = con_specs,

         con_betas = con_betas,

         nchotomy = nchotomy,

         exhaust = exhaust,

         compacts = compacts,

         con_rews = con_rews,

         inverts = inverts,

         injects = injects,

         dist_les = dist_les,

         dist_eqs = dist_eqs,

         cases = cases,

         sel_rews = sel_thms,

         dis_rews = dis_thms,

         match_rews = match_thms

       }

   in

     (result, thy)

   end

 end