(* Title: HOL/Nominal/nominal_primrec.ML
ID: $Id$
Author: Stefan Berghofer, TU Muenchen and Norbert Voelker, FernUni Hagen
Package for defining functions on nominal datatypes by primitive recursion.
Taken from HOL/Tools/primrec_package.ML
*)
signature NOMINAL_PRIMREC =
sig
val add_primrec: string -> string list option -> string option ->
((Name.binding * string) * Attrib.src list) list -> theory -> Proof.state
val add_primrec_unchecked: string -> string list option -> string option ->
((Name.binding * string) * Attrib.src list) list -> theory -> Proof.state
val add_primrec_i: string -> term list option -> term option ->
((Name.binding * term) * attribute list) list -> theory -> Proof.state
val add_primrec_unchecked_i: string -> term list option -> term option ->
((Name.binding * term) * attribute list) list -> theory -> Proof.state
end;
structure NominalPrimrec : NOMINAL_PRIMREC =
struct
open DatatypeAux;
exception RecError of string;
fun primrec_err s = error ("Nominal primrec definition error:\n" ^ s);
fun primrec_eq_err thy s eq =
primrec_err (s ^ "\nin\n" ^ quote (Syntax.string_of_term_global thy eq));
(* preprocessing of equations *)
fun process_eqn thy eq rec_fns =
let
val (lhs, rhs) =
if null (term_vars eq) then
HOLogic.dest_eq (HOLogic.dest_Trueprop (Logic.strip_imp_concl eq))
handle TERM _ => raise RecError "not a proper equation"
else raise RecError "illegal schematic variable(s)";
val (recfun, args) = strip_comb lhs;
val fnameT = dest_Const recfun handle TERM _ =>
raise RecError "function is not declared as constant in theory";
val (ls', rest) = take_prefix is_Free args;
val (middle, rs') = take_suffix is_Free rest;
val rpos = length ls';
val (constr, cargs') = if null middle then raise RecError "constructor missing"
else strip_comb (hd middle);
val (cname, T) = dest_Const constr
handle TERM _ => raise RecError "ill-formed constructor";
val (tname, _) = dest_Type (body_type T) handle TYPE _ =>
raise RecError "cannot determine datatype associated with function"
val (ls, cargs, rs) =
(map dest_Free ls', map dest_Free cargs', map dest_Free rs')
handle TERM _ => raise RecError "illegal argument in pattern";
val lfrees = ls @ rs @ cargs;
fun check_vars _ [] = ()
| check_vars s vars = raise RecError (s ^ commas_quote (map fst vars))
in
if length middle > 1 then
raise RecError "more than one non-variable in pattern"
else
(check_vars "repeated variable names in pattern: " (duplicates (op =) lfrees);
check_vars "extra variables on rhs: "
(map dest_Free (term_frees rhs) \\ lfrees);
case AList.lookup (op =) rec_fns fnameT of
NONE =>
(fnameT, (tname, rpos, [(cname, (ls, cargs, rs, rhs, eq))]))::rec_fns
| SOME (_, rpos', eqns) =>
if AList.defined (op =) eqns cname then
raise RecError "constructor already occurred as pattern"
else if rpos <> rpos' then
raise RecError "position of recursive argument inconsistent"
else
AList.update (op =) (fnameT, (tname, rpos, (cname, (ls, cargs, rs, rhs, eq))::eqns))
rec_fns)
end
handle RecError s => primrec_eq_err thy s eq;
val param_err = "Parameters must be the same for all recursive functions";
fun process_fun thy descr rec_eqns (i, fnameT as (fname, _)) (fnameTs, fnss) =
let
val (_, (tname, _, constrs)) = List.nth (descr, i);
(* substitute "fname ls x rs" by "y" for (x, (_, y)) in subs *)
fun subst [] t fs = (t, fs)
| subst subs (Abs (a, T, t)) fs =
fs
|> subst subs t
|-> (fn t' => pair (Abs (a, T, t')))
| subst subs (t as (_ $ _)) fs =
let
val (f, ts) = strip_comb t;
in
if is_Const f andalso dest_Const f mem map fst rec_eqns then
let
val fnameT' as (fname', _) = dest_Const f;
val (_, rpos, eqns) = the (AList.lookup (op =) rec_eqns fnameT');
val ls = Library.take (rpos, ts);
val rest = Library.drop (rpos, ts);
val (x', rs) = (hd rest, tl rest)
handle Empty => raise RecError ("not enough arguments\
\ in recursive application\nof function " ^ quote fname' ^ " on rhs");
val rs' = (case eqns of
(_, (ls', _, rs', _, _)) :: _ =>
let val (rs1, rs2) = chop (length rs') rs
in
if ls = map Free ls' andalso rs1 = map Free rs' then rs2
else raise RecError param_err
end
| _ => raise RecError ("no equations for " ^ quote fname'));
val (x, xs) = strip_comb x'
in case AList.lookup (op =) subs x
of NONE =>
fs
|> fold_map (subst subs) ts
|-> (fn ts' => pair (list_comb (f, ts')))
| SOME (i', y) =>
fs
|> fold_map (subst subs) (xs @ rs')
||> process_fun thy descr rec_eqns (i', fnameT')
|-> (fn ts' => pair (list_comb (y, ts')))
end
else
fs
|> fold_map (subst subs) (f :: ts)
|-> (fn (f'::ts') => pair (list_comb (f', ts')))
end
| subst _ t fs = (t, fs);
(* translate rec equations into function arguments suitable for rec comb *)
fun trans eqns (cname, cargs) (fnameTs', fnss', fns) =
(case AList.lookup (op =) eqns cname of
NONE => (warning ("No equation for constructor " ^ quote cname ^
"\nin definition of function " ^ quote fname);
(fnameTs', fnss', (Const ("arbitrary", dummyT))::fns))
| SOME (ls, cargs', rs, rhs, eq) =>
let
val recs = filter (is_rec_type o snd) (cargs' ~~ cargs);
val rargs = map fst recs;
val subs = map (rpair dummyT o fst)
(rev (rename_wrt_term rhs rargs));
val (rhs', (fnameTs'', fnss'')) =
(subst (map (fn ((x, y), z) =>
(Free x, (body_index y, Free z)))
(recs ~~ subs)) rhs (fnameTs', fnss'))
handle RecError s => primrec_eq_err thy s eq
in (fnameTs'', fnss'',
(list_abs_free (cargs' @ subs, rhs'))::fns)
end)
in (case AList.lookup (op =) fnameTs i of
NONE =>
if exists (equal fnameT o snd) fnameTs then
raise RecError ("inconsistent functions for datatype " ^ quote tname)
else
let
val SOME (_, _, eqns as (_, (ls, _, rs, _, _)) :: _) =
AList.lookup (op =) rec_eqns fnameT;
val (fnameTs', fnss', fns) = fold_rev (trans eqns) constrs
((i, fnameT)::fnameTs, fnss, [])
in
(fnameTs', (i, (fname, ls, rs, fns))::fnss')
end
| SOME fnameT' =>
if fnameT = fnameT' then (fnameTs, fnss)
else raise RecError ("inconsistent functions for datatype " ^ quote tname))
end;
(* prepare functions needed for definitions *)
fun get_fns fns ((i : int, (tname, _, constrs)), rec_name) (fs, defs) =
case AList.lookup (op =) fns i of
NONE =>
let
val dummy_fns = map (fn (_, cargs) => Const ("arbitrary",
replicate ((length cargs) + (length (List.filter is_rec_type cargs)))
dummyT ---> HOLogic.unitT)) constrs;
val _ = warning ("No function definition for datatype " ^ quote tname)
in
(dummy_fns @ fs, defs)
end
| SOME (fname, ls, rs, fs') => (fs' @ fs, (fname, ls, rs, rec_name, tname) :: defs);
(* make definition *)
fun make_def thy fs (fname, ls, rs, rec_name, tname) =
let
val used = map fst (fold Term.add_frees fs []);
val x = (Name.variant used "x", dummyT);
val frees = ls @ x :: rs;
val rhs = list_abs_free (frees,
list_comb (Const (rec_name, dummyT), fs @ [Free x]))
val def_name = Sign.base_name fname ^ "_" ^ Sign.base_name tname ^ "_def";
val def_prop as _ $ _ $ t =
singleton (Syntax.check_terms (ProofContext.init thy))
(Logic.mk_equals (Const (fname, dummyT), rhs));
in ((def_name, def_prop), subst_bounds (rev (map Free frees), strip_abs_body t)) end;
(* find datatypes which contain all datatypes in tnames' *)
fun find_dts (dt_info : NominalPackage.nominal_datatype_info Symtab.table) _ [] = []
| find_dts dt_info tnames' (tname::tnames) =
(case Symtab.lookup dt_info tname of
NONE => primrec_err (quote tname ^ " is not a nominal datatype")
| SOME dt =>
if tnames' subset (map (#1 o snd) (#descr dt)) then
(tname, dt)::(find_dts dt_info tnames' tnames)
else find_dts dt_info tnames' tnames);
fun common_prefix eq ([], _) = []
| common_prefix eq (_, []) = []
| common_prefix eq (x :: xs, y :: ys) =
if eq (x, y) then x :: common_prefix eq (xs, ys) else [];
local
fun gen_primrec_i note def alt_name invs fctxt eqns_atts thy =
let
val (raw_eqns, atts) = split_list eqns_atts;
val eqns = map (apfst Name.name_of) raw_eqns;
val dt_info = NominalPackage.get_nominal_datatypes thy;
val rec_eqns = fold_rev (process_eqn thy o snd) eqns [];
val lsrs :: lsrss = maps (fn (_, (_, _, eqns)) =>
map (fn (_, (ls, _, rs, _, _)) => ls @ rs) eqns) rec_eqns
val _ =
(if forall (curry eq_set lsrs) lsrss andalso forall
(fn (_, (_, _, (_, (ls, _, rs, _, _)) :: eqns)) =>
forall (fn (_, (ls', _, rs', _, _)) =>
ls = ls' andalso rs = rs') eqns
| _ => true) rec_eqns
then () else primrec_err param_err);
val tnames = distinct (op =) (map (#1 o snd) rec_eqns);
val dts = find_dts dt_info tnames tnames;
val main_fns =
map (fn (tname, {index, ...}) =>
(index,
(fst o the o find_first (fn f => (#1 o snd) f = tname)) rec_eqns))
dts;
val {descr, rec_names, rec_rewrites, ...} =
if null dts then
primrec_err ("datatypes " ^ commas_quote tnames ^ "\nare not mutually recursive")
else snd (hd dts);
val descr = map (fn (i, (tname, args, constrs)) => (i, (tname, args,
map (fn (cname, cargs) => (cname, fold (fn (dTs, dT) => fn dTs' =>
dTs' @ dTs @ [dT]) cargs [])) constrs))) descr;
val (fnameTs, fnss) =
fold_rev (process_fun thy descr rec_eqns) main_fns ([], []);
val (fs, defs) = fold_rev (get_fns fnss) (descr ~~ rec_names) ([], []);
val defs' = map (make_def thy fs) defs;
val nameTs1 = map snd fnameTs;
val nameTs2 = map fst rec_eqns;
val _ = if gen_eq_set (op =) (nameTs1, nameTs2) then ()
else primrec_err ("functions " ^ commas_quote (map fst nameTs2) ^
"\nare not mutually recursive");
val primrec_name =
if alt_name = "" then (space_implode "_" (map (Sign.base_name o #1) defs)) else alt_name;
val (defs_thms', thy') =
thy
|> Sign.add_path primrec_name
|> fold_map def (map (fn ((name, t), _) => ((name, []), t)) defs');
val cert = cterm_of thy';
fun mk_idx eq =
let
val Const c = head_of (fst (HOLogic.dest_eq (HOLogic.dest_Trueprop
(Logic.strip_imp_concl eq))));
val SOME i = AList.lookup op = (map swap fnameTs) c;
val SOME (_, _, constrs) = AList.lookup op = descr i;
val SOME (_, _, eqns) = AList.lookup op = rec_eqns c;
val SOME (cname, (_, cargs, _, _, _)) = find_first
(fn (_, (_, _, _, _, eq')) => eq = eq') eqns
in (i, find_index (fn (cname', _) => cname = cname') constrs, cargs) end;
val rec_rewritess =
unflat (map (fn (_, (_, _, constrs)) => constrs) descr) rec_rewrites;
val fvars = rec_rewrites |> hd |> concl_of |> HOLogic.dest_Trueprop |>
HOLogic.dest_eq |> fst |> strip_comb |> snd |> take_prefix is_Var |> fst;
val (pvars, ctxtvars) = List.partition
(equal HOLogic.boolT o body_type o snd)
(fold_rev Term.add_vars (map Logic.strip_assums_concl
(prems_of (hd rec_rewrites))) [] \\ map dest_Var fvars);
val cfs = defs' |> hd |> snd |> strip_comb |> snd |>
curry (List.take o swap) (length fvars) |> map cert;
val invs' = (case invs of
NONE => map (fn (i, _) =>
let
val SOME (_, T) = AList.lookup op = fnameTs i
val (Ts, U) = strip_type T
in
Abs ("x", List.drop (Ts, length lsrs + 1) ---> U, HOLogic.true_const)
end) descr
| SOME invs' => invs');
val inst = (map cert fvars ~~ cfs) @
(map (cert o Var) pvars ~~ map cert invs') @
(case ctxtvars of
[ctxtvar] => [(cert (Var ctxtvar), cert (the_default HOLogic.unit fctxt))]
| _ => []);
val rec_rewrites' = map (fn (_, eq) =>
let
val (i, j, cargs) = mk_idx eq
val th = nth (nth rec_rewritess i) j;
val cargs' = th |> concl_of |> HOLogic.dest_Trueprop |>
HOLogic.dest_eq |> fst |> strip_comb |> snd |> split_last |> snd |>
strip_comb |> snd
in (cargs, Logic.strip_imp_prems eq,
Drule.cterm_instantiate (inst @
(map (cterm_of thy') cargs' ~~ map (cterm_of thy' o Free) cargs)) th)
end) eqns;
val prems = foldr1 (common_prefix op aconv) (map (prems_of o #3) rec_rewrites');
val cprems = map cert prems;
val asms = map Thm.assume cprems;
val premss = map (fn (cargs, eprems, eqn) =>
map (fn t => list_all_free (cargs, Logic.list_implies (eprems, t)))
(List.drop (prems_of eqn, length prems))) rec_rewrites';
val cpremss = map (map cert) premss;
val asmss = map (map Thm.assume) cpremss;
fun mk_eqn ((cargs, eprems, eqn), asms') =
let
val ceprems = map cert eprems;
val asms'' = map Thm.assume ceprems;
val ccargs = map (cert o Free) cargs;
val asms''' = map (fn th => implies_elim_list
(forall_elim_list ccargs th) asms'') asms'
in
implies_elim_list eqn (asms @ asms''') |>
implies_intr_list ceprems |>
forall_intr_list ccargs
end;
val rule_prems = cprems @ flat cpremss;
val rule = implies_intr_list rule_prems
(Conjunction.intr_balanced (map mk_eqn (rec_rewrites' ~~ asmss)));
val goals = map (fn ((cargs, _, _), (_, eqn)) =>
(list_all_free (cargs, eqn), [])) (rec_rewrites' ~~ eqns);
in
thy' |>
ProofContext.init |>
Proof.theorem_i NONE
(fn thss => ProofContext.theory (fn thy =>
let
val simps = map standard (flat thss);
val (simps', thy') =
fold_map note ((map fst eqns ~~ atts) ~~ map single simps) thy;
val simps'' = maps snd simps'
in
thy'
|> note (("simps", [Simplifier.simp_add]), simps'')
|> snd
|> Sign.parent_path
end))
[goals] |>
Proof.apply (Method.Basic (fn _ => Method.RAW_METHOD (fn _ =>
rewrite_goals_tac (map snd defs_thms') THEN
compose_tac (false, rule, length rule_prems) 1), Position.none)) |>
Seq.hd
end;
fun gen_primrec note def alt_name invs fctxt eqns thy =
let
val ((names, strings), srcss) = apfst split_list (split_list eqns);
val atts = map (map (Attrib.attribute thy)) srcss;
val eqn_ts = map (fn s => Syntax.read_prop_global thy s
handle ERROR msg => cat_error msg ("The error(s) above occurred for " ^ s)) strings;
val rec_ts = map (fn eq => head_of (fst (HOLogic.dest_eq
(HOLogic.dest_Trueprop (Logic.strip_imp_concl eq))))
handle TERM _ => primrec_eq_err thy "not a proper equation" eq) eqn_ts;
val (_, eqn_ts') = OldPrimrecPackage.unify_consts thy rec_ts eqn_ts
in
gen_primrec_i note def alt_name
(Option.map (map (Syntax.read_term_global thy)) invs)
(Option.map (Syntax.read_term_global thy) fctxt)
(names ~~ eqn_ts' ~~ atts) thy
end;
fun thy_note ((name, atts), thms) =
PureThy.add_thmss [((name, thms), atts)] #-> (fn [thms] => pair (name, thms));
fun thy_def false ((name, atts), t) =
PureThy.add_defs false [((name, t), atts)] #-> (fn [thm] => pair (name, thm))
| thy_def true ((name, atts), t) =
PureThy.add_defs_unchecked false [((name, t), atts)] #-> (fn [thm] => pair (name, thm));
in
val add_primrec = gen_primrec thy_note (thy_def false);
val add_primrec_unchecked = gen_primrec thy_note (thy_def true);
val add_primrec_i = gen_primrec_i thy_note (thy_def false);
val add_primrec_unchecked_i = gen_primrec_i thy_note (thy_def true);
end; (*local*)
(* outer syntax *)
local structure P = OuterParse and K = OuterKeyword in
val freshness_context = P.reserved "freshness_context";
val invariant = P.reserved "invariant";
fun unless_flag scan = Scan.unless ((freshness_context || invariant) -- P.$$$ ":") scan;
val parser1 = (freshness_context -- P.$$$ ":") |-- unless_flag P.term >> SOME;
val parser2 = (invariant -- P.$$$ ":") |--
(Scan.repeat1 (unless_flag P.term) >> SOME) -- Scan.optional parser1 NONE ||
(parser1 >> pair NONE);
val parser3 =
unless_flag P.name -- Scan.optional parser2 (NONE, NONE) ||
(parser2 >> pair "");
val parser4 =
(P.$$$ "unchecked" >> K true) -- Scan.optional parser3 ("", (NONE, NONE)) ||
(parser3 >> pair false);
val options =
Scan.optional (P.$$$ "(" |-- P.!!!
(parser4 --| P.$$$ ")")) (false, ("", (NONE, NONE)));
val primrec_decl =
options -- Scan.repeat1 (SpecParse.opt_thm_name ":" -- P.prop);
val _ =
OuterSyntax.command "nominal_primrec" "define primitive recursive functions on nominal datatypes" K.thy_goal
(primrec_decl >> (fn ((unchecked, (alt_name, (invs, fctxt))), eqns) =>
Toplevel.print o Toplevel.theory_to_proof
((if unchecked then add_primrec_unchecked else add_primrec) alt_name invs fctxt
(map P.triple_swap eqns))));
end;
end;