(* 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_package.ML *) signature NOMINAL_PRIMREC = sig val add_primrec: term list option -> term option -> (binding * typ option * mixfix) list -> (binding * typ option * mixfix) list -> (Attrib.binding * term) list -> local_theory -> Proof.state val add_primrec_cmd: string list option -> string option -> (binding * string option * mixfix) list -> (binding * string option * mixfix) list -> (Attrib.binding * string) list -> local_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 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 "all" t); val body = strip_qnt_body "all" t; val (vs', _) = Name.variants 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) = 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 (OldTerm.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 (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, (body_index y, Free z))) recs subs) rhs (fnames', fnss') handle RecError s => primrec_eq_err lthy s eq in (fnames'', fnss'', (list_abs_free (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 (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 ctxt fixes 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 raw_rhs = list_abs_free (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 : 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 set_group 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 = NominalPackage.get_nominal_datatypes (ProofContext.theory_of lthy); val lsrs :: lsrss = maps (fn (_, (_, _, eqns)) => map (fn (_, (ls, _, rs, _, _)) => ls @ rs) eqns) 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) 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 gen_eq_set (op =) (names1, names2) then () else primrec_err ("functions " ^ commas_quote names2 ^ "\nare not mutually recursive"); val (defs_thms, lthy') = lthy |> set_group ? LocalTheory.set_group (serial_string ()) |> fold_map (apfst (snd o snd) oo LocalTheory.define Thm.definitionK 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; val cert = cterm_of (ProofContext.theory_of lthy'); 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 |> 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, _) => Abs ("x", fastype_of (snd (nth defs' i)), HOLogic.true_const)) descr | SOME invs' => map (prep_term lthy') 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 (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 |> 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 cert cargs' ~~ map (cert 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 lthy' |> Variable.add_fixes (map fst lsrs) |> snd |> Proof.theorem_i NONE (fn thss => fn goal_ctxt => let val simps = ProofContext.export goal_ctxt lthy' (flat thss); val (simps', lthy'') = fold_map (LocalTheory.note Thm.theoremK) (names_atts' ~~ map single simps) lthy' in lthy'' |> LocalTheory.note Thm.theoremK ((qualify (Binding.name "simps"), map (Attrib.internal o K) [Simplifier.simp_add, Nitpick_Const_Simp_Thms.add]), maps snd simps') |> snd end) [goals] |> Proof.apply (Method.Basic (fn _ => RAW_METHOD (fn _ => rewrite_goals_tac defs_thms THEN compose_tac (false, rule, length rule_prems) 1), Position.none)) |> Seq.hd end; in val add_primrec = gen_primrec false Specification.check_spec (K I); val add_primrec_cmd = gen_primrec true Specification.read_spec Syntax.read_term; end; (* outer syntax *) local structure P = OuterParse 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.$$$ ":") |-- (Scan.repeat1 (unless_flag P.term) >> SOME) -- Scan.optional parser1 NONE || (parser1 >> pair NONE); val options = Scan.optional (P.$$$ "(" |-- P.!!! (parser2 --| P.$$$ ")")) (NONE, NONE); val _ = OuterSyntax.local_theory_to_proof "nominal_primrec" "define primitive recursive functions on nominal datatypes" OuterKeyword.thy_goal (options -- P.fixes -- P.for_fixes -- SpecParse.where_alt_specs >> (fn ((((invs, fctxt), fixes), params), specs) => add_primrec_cmd invs fctxt fixes params specs)); end; end;