(* Title: 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
val add_domain_constructors :
binding
-> (binding * (bool * binding option * typ) list * mixfix) list
-> Domain_Take_Proofs.iso_info
-> theory
-> { con_consts : term 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,
pat_rews : thm list
} * theory;
end;
structure Domain_Constructors :> DOMAIN_CONSTRUCTORS =
struct
open HOLCF_Library;
infixr 6 ->>;
infix -->>;
infix 9 `;
(************************** miscellaneous functions ***************************)
val simple_ss = HOL_basic_ss addsimps simp_thms;
val beta_rules =
@{thms beta_cfun cont_id cont_const cont2cont_Rep_CFun 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) =
PureThy.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_defined_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 = UU \/ P" with "rep$y = UU \/ 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 UU = mk_bottom (fastype_of v');
val vs' = map (fn v => if v = v' then UU 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_defined_iff};
val rules = rule1 :: @{thms con_defined_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 o dest_TFree) (snd (dest_Type lhsT));
val resultT = TFree (Name.variant 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 "_when" dbind;
fun lambda_arg (lazy, v) t =
(if lazy then mk_fup else I) (big_lambda v t);
fun lambda_args [] t = mk_one_when 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
open Syntax
fun syntax c = Syntax.mark_const (fst (dest_Const c));
fun xconst c = Long_Name.base_name (fst (dest_Const c));
fun c_ast authentic con =
Constant (if authentic then syntax con else xconst con);
fun showint n = string_of_int (n+1);
fun expvar n = Variable ("e" ^ showint n);
fun argvar n (m, _) = Variable ("a" ^ showint n ^ "_" ^ showint m);
fun argvars n args = map_index (argvar n) args;
fun app s (l, r) = mk_appl (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" (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 (Constant @{const_syntax UU});
val case_constant = Constant (syntax (case_const dummyT));
fun case_trans authentic =
ParsePrintRule
(app "_case_syntax"
(Variable "x",
foldr1 (app "_case2") (map_index (case1 authentic) spec)),
capp (capps (case_constant, map_index arg1 spec), Variable "x"));
fun one_abscon_trans authentic (n, c) =
ParsePrintRule
(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 Syntax.trrule list =
case_trans false :: case_trans true ::
abscon_trans false @ abscon_trans true;
in
val thy = Sign.add_trrules_i 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_when1};
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_defined_iff_rules};
val rules3 = @{thms cfcomp2 one_when2};
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_strict_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)) =
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_strict_iff :: @{thms sel_defined_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 o dest_TFree) (snd (dest_Type lhsT));
val t : string = Name.variant 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;
(******************************************************************************)
(************** definitions and theorems for pattern combinators **************)
(******************************************************************************)
fun add_pattern_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
(* utility functions *)
fun mk_pair_pat (p1, p2) =
let
val T1 = fastype_of p1;
val T2 = fastype_of p2;
val (U1, V1) = apsnd dest_matchT (dest_cfunT T1);
val (U2, V2) = apsnd dest_matchT (dest_cfunT T2);
val pat_typ = [T1, T2] --->
(mk_prodT (U1, U2) ->> mk_matchT (mk_prodT (V1, V2)));
val pat_const = Const (@{const_name cpair_pat}, pat_typ);
in
pat_const $ p1 $ p2
end;
fun mk_tuple_pat [] = succeed_const HOLogic.unitT
| mk_tuple_pat ps = foldr1 mk_pair_pat ps;
fun branch_const (T,U,V) =
Const (@{const_name branch},
(T ->> mk_matchT U) --> (U ->> V) ->> T ->> mk_matchT V);
(* define pattern combinators *)
local
val tns = map (fst o dest_TFree) (snd (dest_Type lhsT));
fun pat_eqn (i, (bind, (con, args))) : binding * term * mixfix =
let
val pat_bind = Binding.suffix_name "_pat" bind;
val Ts = map snd args;
val Vs =
(map (K "'t") args)
|> Datatype_Prop.indexify_names
|> Name.variant_list tns
|> map (fn t => TFree (t, @{sort pcpo}));
val patNs = Datatype_Prop.indexify_names (map (K "pat") args);
val patTs = map2 (fn T => fn V => T ->> mk_matchT V) Ts Vs;
val pats = map Free (patNs ~~ patTs);
val fail = mk_fail (mk_tupleT Vs);
val (vs, nonlazy) = get_vars_avoiding patNs args;
val rhs = big_lambdas vs (mk_tuple_pat pats ` mk_tuple vs);
fun one_fun (j, (_, args')) =
let
val (vs', nonlazy) = get_vars_avoiding patNs args';
in if i = j then rhs else big_lambdas vs' fail end;
val funs = map_index one_fun spec;
val body = list_ccomb (case_const (mk_matchT (mk_tupleT Vs)), funs);
in
(pat_bind, lambdas pats body, NoSyn)
end;
in
val ((pat_consts, pat_defs), thy) =
define_consts (map_index pat_eqn (bindings ~~ spec)) thy
end;
(* syntax translations for pattern combinators *)
local
open Syntax
fun syntax c = Syntax.mark_const (fst (dest_Const c));
fun app s (l, r) = Syntax.mk_appl (Constant s) [l, r];
val capp = app @{const_syntax Rep_CFun};
val capps = Library.foldl capp
fun app_var x = Syntax.mk_appl (Constant "_variable") [x, Variable "rhs"];
fun app_pat x = Syntax.mk_appl (Constant "_pat") [x];
fun args_list [] = Constant "_noargs"
| args_list xs = foldr1 (app "_args") xs;
fun one_case_trans (pat, (con, args)) =
let
val cname = Constant (syntax con);
val pname = Constant (syntax pat);
val ns = 1 upto length args;
val xs = map (fn n => Variable ("x"^(string_of_int n))) ns;
val ps = map (fn n => Variable ("p"^(string_of_int n))) ns;
val vs = map (fn n => Variable ("v"^(string_of_int n))) ns;
in
[ParseRule (app_pat (capps (cname, xs)),
mk_appl pname (map app_pat xs)),
ParseRule (app_var (capps (cname, xs)),
app_var (args_list xs)),
PrintRule (capps (cname, ListPair.map (app "_match") (ps,vs)),
app "_match" (mk_appl pname ps, args_list vs))]
end;
val trans_rules : Syntax.ast Syntax.trrule list =
maps one_case_trans (pat_consts ~~ spec);
in
val thy = Sign.add_trrules_i trans_rules thy;
end;
(* prove strictness and reduction rules of pattern combinators *)
local
val tns = map (fst o dest_TFree) (snd (dest_Type lhsT));
val rn = Name.variant tns "'r";
val R = TFree (rn, @{sort pcpo});
fun pat_lhs (pat, args) =
let
val Ts = map snd args;
val Vs =
(map (K "'t") args)
|> Datatype_Prop.indexify_names
|> Name.variant_list (rn::tns)
|> map (fn t => TFree (t, @{sort pcpo}));
val patNs = Datatype_Prop.indexify_names (map (K "pat") args);
val patTs = map2 (fn T => fn V => T ->> mk_matchT V) Ts Vs;
val pats = map Free (patNs ~~ patTs);
val k = Free ("rhs", mk_tupleT Vs ->> R);
val branch1 = branch_const (lhsT, mk_tupleT Vs, R);
val fun1 = (branch1 $ list_comb (pat, pats)) ` k;
val branch2 = branch_const (mk_tupleT Ts, mk_tupleT Vs, R);
val fun2 = (branch2 $ mk_tuple_pat pats) ` k;
val taken = "rhs" :: patNs;
in (fun1, fun2, taken) end;
fun pat_strict (pat, (con, args)) =
let
val (fun1, fun2, taken) = pat_lhs (pat, args);
val defs = @{thm branch_def} :: pat_defs;
val goal = mk_trp (mk_strict fun1);
val rules = @{thms match_case_simps} @ case_rews;
val tacs = [simp_tac (beta_ss addsimps rules) 1];
in prove thy defs goal (K tacs) end;
fun pat_apps (i, (pat, (con, args))) =
let
val (fun1, fun2, taken) = pat_lhs (pat, args);
fun pat_app (j, (con', args')) =
let
val (vs, nonlazy) = get_vars_avoiding taken args';
val con_app = list_ccomb (con', vs);
val assms = map (mk_trp o mk_defined) nonlazy;
val rhs = if i = j then fun2 ` mk_tuple vs else mk_fail R;
val concl = mk_trp (mk_eq (fun1 ` con_app, rhs));
val goal = Logic.list_implies (assms, concl);
val defs = @{thm branch_def} :: pat_defs;
val rules = @{thms match_case_simps} @ case_rews;
val tacs = [asm_simp_tac (beta_ss addsimps rules) 1];
in prove thy defs goal (K tacs) end;
in map_index pat_app spec end;
in
val pat_stricts = map pat_strict (pat_consts ~~ spec);
val pat_apps = flat (map_index pat_apps (pat_consts ~~ spec));
end;
in
(pat_stricts @ pat_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;
(* 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_defined_iff = iso_locale RS @{thm iso.rep_defined_iff};
val abs_defined_iff = iso_locale RS @{thm iso.abs_defined_iff};
(* 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, exhaust, ...} = con_result;
(* define case combinator *)
val ((case_const : typ -> term, cases : thm list), thy) =
let
fun prep_arg (lazy, sel, T) = (lazy, T);
fun prep_con c (b, args, mx) = (c, map prep_arg args);
val case_spec = map2 prep_con con_consts spec;
in
add_case_combinator case_spec lhsT dbind
con_betas exhaust iso_locale rep_const thy
end;
(* 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_defined_iff con_betas thy
end;
(* define and prove theorems for discriminator functions *)
val (dis_thms : thm list, thy : theory) =
let
val bindings = map #1 spec;
fun prep_arg (lazy, sel, T) = (lazy, T);
fun prep_con c (b, args, mx) = (c, map prep_arg args);
val dis_spec = map2 prep_con con_consts spec;
in
add_discriminators bindings dis_spec lhsT
exhaust case_const cases thy
end
(* define and prove theorems for match combinators *)
val (match_thms : thm list, thy : theory) =
let
val bindings = map #1 spec;
fun prep_arg (lazy, sel, T) = (lazy, T);
fun prep_con c (b, args, mx) = (c, map prep_arg args);
val mat_spec = map2 prep_con con_consts spec;
in
add_match_combinators bindings mat_spec lhsT
exhaust case_const cases thy
end
(* define and prove theorems for pattern combinators *)
val (pat_thms : thm list, thy : theory) =
let
val bindings = map #1 spec;
fun prep_arg (lazy, sel, T) = (lazy, T);
fun prep_con c (b, args, mx) = (c, map prep_arg args);
val pat_spec = map2 prep_con con_consts spec;
in
add_pattern_combinators bindings pat_spec lhsT
exhaust case_const cases thy
end
(* restore original signature path *)
val thy = Sign.parent_path thy;
val result =
{ con_consts = con_consts,
con_betas = con_betas,
nchotomy = #nchotomy con_result,
exhaust = exhaust,
compacts = #compacts con_result,
con_rews = #con_rews con_result,
inverts = #inverts con_result,
injects = #injects con_result,
dist_les = #dist_les con_result,
dist_eqs = #dist_eqs con_result,
cases = cases,
sel_rews = sel_thms,
dis_rews = dis_thms,
match_rews = match_thms,
pat_rews = pat_thms };
in
(result, thy)
end;
end;