(* Title: HOL/Nominal/nominal_primrec.ML ID: $Id: nominal_primrec.ML,v 1.18 2007/10/08 16:13:05 wenzelm Exp $ 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 quiet_mode: bool ref val add_primrec: string -> string list option -> string option -> ((bstring * string) * Attrib.src list) list -> theory -> Proof.state val add_primrec_unchecked: string -> string list option -> string option -> ((bstring * string) * Attrib.src list) list -> theory -> Proof.state val add_primrec_i: string -> term list option -> term option -> ((bstring * term) * attribute list) list -> theory -> Proof.state val add_primrec_unchecked_i: string -> term list option -> term option -> ((bstring * 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 (Sign.string_of_term thy eq)); (* messages *) val quiet_mode = ref false; fun message s = if ! quiet_mode then () else writeln s; (* 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 (eqns, atts) = split_list eqns_atts; 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') = PrimrecPackage.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_i false [((name, t), atts)] #-> (fn [thm] => pair (name, thm)) | thy_def true ((name, atts), t) = PureThy.add_defs_unchecked_i 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;