(* Title: HOL/Nominal/nominal_primrec.ML
Author: Norbert Voelker, FernUni Hagen
Author: Stefan Berghofer, TU Muenchen
Package for defining functions on nominal datatypes by primitive recursion.
Taken from HOL/Tools/primrec.ML
*)
signature NOMINAL_PRIMREC =
sig
val primrec: term list option -> term option ->
(binding * typ option * mixfix) list ->
(binding * typ option * mixfix) list ->
Specification.multi_specs -> local_theory -> Proof.state
val primrec_cmd: string list option -> string option ->
(binding * string option * mixfix) list ->
(binding * string option * mixfix) list ->
Specification.multi_specs_cmd -> local_theory -> Proof.state
end;
structure NominalPrimrec : NOMINAL_PRIMREC =
struct
exception RecError of string;
fun primrec_err s = error ("Nominal primrec definition error:\n" ^ s);
fun primrec_eq_err lthy s eq =
primrec_err (s ^ "\nin\n" ^ quote (Syntax.string_of_term lthy eq));
(* preprocessing of equations *)
fun unquantify t =
let
val (vs, Ts) = split_list (strip_qnt_vars @{const_name Pure.all} t);
val body = strip_qnt_body @{const_name Pure.all} t;
val (vs', _) = fold_map Name.variant vs (Name.make_context (fold_aterms
(fn Free (v, _) => insert (op =) v | _ => I) body []))
in (curry subst_bounds (map2 (curry Free) vs' Ts |> rev) body) end;
fun process_eqn lthy is_fixed spec rec_fns =
let
val eq = unquantify spec;
val (lhs, rhs) =
HOLogic.dest_eq (HOLogic.dest_Trueprop (Logic.strip_imp_concl eq))
handle TERM _ => raise RecError "not a proper equation";
val (recfun, args) = strip_comb lhs;
val fname = case recfun of Free (v, _) => if is_fixed v then v
else raise RecError "illegal head of function equation"
| _ => raise RecError "illegal head of function equation";
val (ls', rest) = chop_prefix is_Free args;
val (middle, rs') = chop_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 (Misc_Legacy.term_frees rhs) |> subtract (op =) lfrees
|> filter_out (is_fixed o fst));
case AList.lookup (op =) rec_fns fname of
NONE =>
(fname, (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 =)
(fname, (tname, rpos, (cname, (ls, cargs, rs, rhs, eq))::eqns))
rec_fns)
end
handle RecError s => primrec_eq_err lthy s spec;
val param_err = "Parameters must be the same for all recursive functions";
fun process_fun lthy descr eqns (i, fname) (fnames, fnss) =
let
val (_, (tname, _, constrs)) = 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_Free f
andalso member (fn ((v, _), (w, _)) => v = w) eqns (dest_Free f) then
let
val (fname', _) = dest_Free f;
val (_, rpos, eqns') = the (AList.lookup (op =) eqns fname');
val (ls, rs'') = chop rpos ts
val (x', rs) = case rs'' of
x' :: rs => (x', rs)
| [] => raise RecError ("not enough arguments in recursive application\n"
^ "of 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 lthy descr eqns (i', fname')
|-> (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) (fnames', fnss', fns) =
(case AList.lookup (op =) eqns cname of
NONE => (warning ("No equation for constructor " ^ quote cname ^
"\nin definition of function " ^ quote fname);
(fnames', fnss', (Const (@{const_name undefined}, dummyT))::fns))
| SOME (ls, cargs', rs, rhs, eq) =>
let
val recs = filter (Old_Datatype_Aux.is_rec_type o snd) (cargs' ~~ cargs);
val rargs = map fst recs;
val subs = map (rpair dummyT o fst)
(rev (Term.rename_wrt_term rhs rargs));
val (rhs', (fnames'', fnss'')) = subst (map2 (fn (x, y) => fn z =>
(Free x, (Old_Datatype_Aux.body_index y, Free z))) recs subs) rhs (fnames', fnss')
handle RecError s => primrec_eq_err lthy s eq
in (fnames'', fnss'', fold_rev absfree (cargs' @ subs) rhs' :: fns)
end)
in (case AList.lookup (op =) fnames i of
NONE =>
if exists (fn (_, v) => fname = v) fnames then
raise RecError ("inconsistent functions for datatype " ^ quote tname)
else
let
val SOME (_, _, eqns' as (_, (ls, _, rs, _, _)) :: _) =
AList.lookup (op =) eqns fname;
val (fnames', fnss', fns) = fold_rev (trans eqns') constrs
((i, fname)::fnames, fnss, [])
in
(fnames', (i, (fname, ls, rs, fns))::fnss')
end
| SOME fname' =>
if fname = fname' then (fnames, 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 (@{const_name undefined},
replicate (length cargs + length (filter Old_Datatype_Aux.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 ctxt fixes fs (fname, ls, rs, rec_name, tname) =
let
val used = map fst (fold Term.add_frees fs []);
val x = (singleton (Name.variant_list used) "x", dummyT);
val frees = ls @ x :: rs;
val raw_rhs = fold_rev absfree frees
(list_comb (Const (rec_name, dummyT), fs @ [Free x]))
val def_name = Thm.def_name (Long_Name.base_name fname);
val rhs = singleton (Syntax.check_terms ctxt) raw_rhs;
val SOME var = get_first (fn ((b, _), mx) =>
if Binding.name_of b = fname then SOME (b, mx) else NONE) fixes;
in
((var, ((Binding.name def_name, []), rhs)),
subst_bounds (rev (map Free frees), strip_abs_body rhs))
end;
(* find datatypes which contain all datatypes in tnames' *)
fun find_dts (dt_info : NominalDatatype.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 subset (op =) (tnames', 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 prep_spec prep_term invs fctxt raw_fixes raw_params raw_spec lthy =
let
val (fixes', spec) = fst (prep_spec (raw_fixes @ raw_params) raw_spec lthy);
val fixes = List.take (fixes', length raw_fixes);
val (names_atts, spec') = split_list spec;
val eqns' = map unquantify spec'
val eqns = fold_rev (process_eqn lthy (fn v => Variable.is_fixed lthy v
orelse exists (fn ((w, _), _) => v = Binding.name_of w) fixes)) spec' [];
val dt_info = NominalDatatype.get_nominal_datatypes (Proof_Context.theory_of lthy);
val lsrs :: lsrss = maps (fn (_, (_, _, eqns)) =>
map (fn (_, (ls, _, rs, _, _)) => ls @ rs) eqns) eqns
val _ =
(if forall (curry (eq_set (op =)) lsrs) lsrss andalso forall
(fn (_, (_, _, (_, (ls, _, rs, _, _)) :: eqns)) =>
forall (fn (_, (ls', _, rs', _, _)) =>
ls = ls' andalso rs = rs') eqns
| _ => true) eqns
then () else primrec_err param_err);
val tnames = distinct (op =) (map (#1 o snd) eqns);
val dts = find_dts dt_info tnames tnames;
val main_fns =
map (fn (tname, {index, ...}) =>
(index,
(fst o the o find_first (fn (_, x) => #1 x = tname)) 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 (fnames, fnss) = fold_rev (process_fun lthy descr eqns) main_fns ([], []);
val (fs, defs) = fold_rev (get_fns fnss) (descr ~~ rec_names) ([], []);
val defs' = map (make_def lthy fixes fs) defs;
val names1 = map snd fnames;
val names2 = map fst eqns;
val _ = if eq_set (op =) (names1, names2) then ()
else primrec_err ("functions " ^ commas_quote names2 ^
"\nare not mutually recursive");
val (defs_thms, lthy') = lthy |>
fold_map (apfst (snd o snd) oo Local_Theory.define o fst) defs';
val qualify = Binding.qualify false
(space_implode "_" (map (Long_Name.base_name o #1) defs));
val names_atts' = map (apfst qualify) names_atts;
fun mk_idx eq =
let
val Free (name, _) = head_of (fst (HOLogic.dest_eq (HOLogic.dest_Trueprop
(Logic.strip_imp_concl eq))));
val SOME i = AList.lookup op = (map swap fnames) name;
val SOME (_, _, constrs) = AList.lookup op = descr i;
val SOME (_, _, eqns'') = AList.lookup op = eqns name;
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 |> Thm.concl_of |> HOLogic.dest_Trueprop |>
HOLogic.dest_eq |> fst |> strip_comb |> snd |> take_prefix is_Var;
val (pvars, ctxtvars) = List.partition
(equal HOLogic.boolT o body_type o snd)
(subtract (op =)
(Term.add_vars (Thm.concl_of (hd rec_rewrites)) [])
(fold_rev (Term.add_vars o Logic.strip_assums_concl)
(Thm.prems_of (hd rec_rewrites)) []));
val cfs = defs' |> hd |> snd |> strip_comb |> snd |>
curry (List.take o swap) (length fvars) |> map (Thm.cterm_of lthy');
val invs' = (case invs of
NONE => map (fn (i, _) =>
Abs ("x", fastype_of (snd (nth defs' i)), @{term True})) descr
| SOME invs' => map (prep_term lthy') invs');
val inst = (map (#1 o dest_Var) fvars ~~ cfs) @
(map #1 pvars ~~ map (Thm.cterm_of lthy') invs') @
(case ctxtvars of
[ctxtvar] => [(#1 ctxtvar,
Thm.cterm_of lthy' (the_default HOLogic.unit (Option.map (prep_term lthy') 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 |> Thm.concl_of |> HOLogic.dest_Trueprop |>
HOLogic.dest_eq |> fst |> strip_comb |> snd |> List.last |>
strip_comb |> snd
in (cargs, Logic.strip_imp_prems eq,
infer_instantiate lthy' (inst @
(map (#1 o dest_Var) cargs' ~~ map (Thm.cterm_of lthy' o Free) cargs)) th)
end) eqns';
val prems = foldr1 (common_prefix op aconv) (map (Thm.prems_of o #3) rec_rewrites');
val cprems = map (Thm.cterm_of lthy') prems;
val asms = map Thm.assume cprems;
val premss = map (fn (cargs, eprems, eqn) =>
map (fn t => fold_rev (Logic.all o Free) cargs (Logic.list_implies (eprems, t)))
(List.drop (Thm.prems_of eqn, length prems))) rec_rewrites';
val cpremss = map (map (Thm.cterm_of lthy')) premss;
val asmss = map (map Thm.assume) cpremss;
fun mk_eqn ((cargs, eprems, eqn), asms') =
let
val ceprems = map (Thm.cterm_of lthy') eprems;
val asms'' = map Thm.assume ceprems;
val ccargs = map (Thm.cterm_of lthy' 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) =>
(fold_rev (Logic.all o Free) cargs eqn, [])) (rec_rewrites' ~~ eqns');
in
lthy' |>
Variable.add_fixes (map fst lsrs) |> snd |>
Proof.theorem NONE
(fn thss => fn goal_ctxt =>
let
val simps = Proof_Context.export goal_ctxt lthy' (flat thss);
val (simps', lthy'') =
fold_map Local_Theory.note (names_atts' ~~ map single simps) lthy';
in
lthy''
|> Local_Theory.note
((qualify (Binding.name "simps"), @{attributes [simp, nitpick_simp]}), maps snd simps')
|> snd
end)
[goals] |>
Proof.refine_singleton (Method.Basic (fn ctxt => fn _ =>
Method.CONTEXT_TACTIC
(rewrite_goals_tac ctxt defs_thms THEN
compose_tac ctxt (false, rule, length rule_prems) 1)))
end;
in
val primrec = gen_primrec Specification.check_multi_specs (K I);
val primrec_cmd = gen_primrec Specification.read_multi_specs Syntax.read_term;
end;
(* outer syntax *)
val freshness_context = Parse.reserved "freshness_context";
val invariant = Parse.reserved "invariant";
fun unless_flag scan = Scan.unless ((freshness_context || invariant) -- @{keyword ":"}) scan;
val parser1 = (freshness_context -- @{keyword ":"}) |-- unless_flag Parse.term >> SOME;
val parser2 = (invariant -- @{keyword ":"}) |--
(Scan.repeat1 (unless_flag Parse.term) >> SOME) -- Scan.optional parser1 NONE ||
(parser1 >> pair NONE);
val options =
Scan.optional (@{keyword "("} |-- Parse.!!! (parser2 --| @{keyword ")"})) (NONE, NONE);
val _ =
Outer_Syntax.local_theory_to_proof @{command_keyword nominal_primrec}
"define primitive recursive functions on nominal datatypes"
(options -- Parse.vars -- Parse.for_fixes -- Parse_Spec.where_multi_specs
>> (fn ((((invs, fctxt), vars), params), specs) =>
primrec_cmd invs fctxt vars params specs));
end;