(* Title: Tools/nbe.ML Authors: Klaus Aehlig, LMU Muenchen; Tobias Nipkow, Florian Haftmann, TU MuenchenNormalization by evaluation, based on generic code generator.*)signature NBE =sig val norm_conv: cterm -> thm val norm: theory -> term -> term datatype Univ = Const of int * Univ list (*named (uninterpreted) constants*) | DFree of string * int (*free (uninterpreted) dictionary parameters*) | BVar of int * Univ list | Abs of (int * (Univ list -> Univ)) * Univ list val apps: Univ -> Univ list -> Univ (*explicit applications*) val abss: int -> (Univ list -> Univ) -> Univ (*abstractions as closures*) val same: Univ -> Univ -> bool val univs_ref: (unit -> Univ list -> Univ list) option ref val trace: bool ref val setup: theory -> theoryend;structure Nbe: NBE =struct(* generic non-sense *)val trace = ref false;fun tracing f x = if !trace then (Output.tracing (f x); x) else x;(** the semantical universe **)(* Functions are given by their semantical function value. To avoid trouble with the ML-type system, these functions have the most generic type, that is "Univ list -> Univ". The calling convention is that the arguments come as a list, the last argument first. In other words, a function call that usually would look like f x_1 x_2 ... x_n or f(x_1,x_2, ..., x_n) would be in our convention called as f [x_n,..,x_2,x_1] Moreover, to handle functions that are still waiting for some arguments we have additionally a list of arguments collected to far and the number of arguments we're still waiting for.*)datatype Univ = Const of int * Univ list (*named (uninterpreted) constants*) | DFree of string * int (*free (uninterpreted) dictionary parameters*) | BVar of int * Univ list (*bound variables, named*) | Abs of (int * (Univ list -> Univ)) * Univ list (*abstractions as closures*);fun same (Const (k, xs)) (Const (l, ys)) = k = l andalso sames xs ys | same (DFree (s, k)) (DFree (t, l)) = s = t andalso k = l | same (BVar (k, xs)) (BVar (l, ys)) = k = l andalso sames xs ys | same _ _ = falseand sames xs ys = length xs = length ys andalso forall (uncurry same) (xs ~~ ys);(* constructor functions *)fun abss n f = Abs ((n, f), []);fun apps (Abs ((n, f), xs)) ys = let val k = n - length ys in case int_ord (k, 0) of EQUAL => f (ys @ xs) | LESS => let val (zs, ws) = chop (~ k) ys in apps (f (ws @ xs)) zs end | GREATER => Abs ((k, f), ys @ xs) (*note: reverse convention also for apps!*) end | apps (Const (name, xs)) ys = Const (name, ys @ xs) | apps (BVar (n, xs)) ys = BVar (n, ys @ xs);(** assembling and compiling ML code from terms **)(* abstract ML syntax *)infix 9 `$` `$$`;fun e1 `$` e2 = "(" ^ e1 ^ " " ^ e2 ^ ")";fun e `$$` [] = e | e `$$` es = "(" ^ e ^ " " ^ space_implode " " es ^ ")";fun ml_abs v e = "(fn " ^ v ^ " => " ^ e ^ ")";fun ml_cases t cs = "(case " ^ t ^ " of " ^ space_implode " | " (map (fn (p, t) => p ^ " => " ^ t) cs) ^ ")";fun ml_Let d e = "let\n" ^ d ^ " in " ^ e ^ " end";fun ml_as v t = "(" ^ v ^ " as " ^ t ^ ")";fun ml_and [] = "true" | ml_and [x] = x | ml_and xs = "(" ^ space_implode " andalso " xs ^ ")";fun ml_if b x y = "(if " ^ b ^ " then " ^ x ^ " else " ^ y ^ ")";fun ml_list es = "[" ^ commas es ^ "]";fun ml_fundefs ([(name, [([], e)])]) = "val " ^ name ^ " = " ^ e ^ "\n" | ml_fundefs (eqs :: eqss) = let fun fundef (name, eqs) = let fun eqn (es, e) = name ^ " " ^ space_implode " " es ^ " = " ^ e in space_implode "\n | " (map eqn eqs) end; in (prefix "fun " o fundef) eqs :: map (prefix "and " o fundef) eqss |> cat_lines |> suffix "\n" end;(* nbe specific syntax and sandbox communication *)val univs_ref = ref (NONE : (unit -> Univ list -> Univ list) option);local val prefix = "Nbe."; val name_ref = prefix ^ "univs_ref"; val name_const = prefix ^ "Const"; val name_abss = prefix ^ "abss"; val name_apps = prefix ^ "apps"; val name_same = prefix ^ "same";inval univs_cookie = (name_ref, univs_ref);fun nbe_fun 0 "" = "nbe_value" | nbe_fun i c = "c_" ^ translate_string (fn "." => "_" | c => c) c ^ "_" ^ string_of_int i;fun nbe_dict v n = "d_" ^ v ^ "_" ^ string_of_int n;fun nbe_bound v = "v_" ^ v;fun nbe_default v = "w_" ^ v;(*note: these three are the "turning spots" where proper argument order is established!*)fun nbe_apps t [] = t | nbe_apps t ts = name_apps `$$` [t, ml_list (rev ts)];fun nbe_apps_local i c ts = nbe_fun i c `$` ml_list (rev ts);fun nbe_apps_constr idx_of c ts = let val c' = if !trace then string_of_int (idx_of c) ^ " (*" ^ c ^ "*)" else string_of_int (idx_of c); in name_const `$` ("(" ^ c' ^ ", " ^ ml_list (rev ts) ^ ")") end;fun nbe_abss 0 f = f `$` ml_list [] | nbe_abss n f = name_abss `$$` [string_of_int n, f];fun nbe_same v1 v2 = "(" ^ name_same ^ " " ^ nbe_bound v1 ^ " " ^ nbe_bound v2 ^ ")";end;open Basic_Code_Thingol;(* code generation *)fun assemble_eqnss idx_of deps eqnss = let fun prep_eqns (c, (vs, eqns)) = let val dicts = maps (fn (v, sort) => map_index (nbe_dict v o fst) sort) vs; val num_args = length dicts + ((length o fst o hd) eqns); in (c, (num_args, (dicts, eqns))) end; val eqnss' = map prep_eqns eqnss; fun assemble_constapp c dss ts = let val ts' = (maps o map) assemble_idict dss @ ts; in case AList.lookup (op =) eqnss' c of SOME (num_args, _) => if num_args <= length ts' then let val (ts1, ts2) = chop num_args ts' in nbe_apps (nbe_apps_local 0 c ts1) ts2 end else nbe_apps (nbe_abss num_args (nbe_fun 0 c)) ts' | NONE => if member (op =) deps c then nbe_apps (nbe_fun 0 c) ts' else nbe_apps_constr idx_of c ts' end and assemble_idict (DictConst (inst, dss)) = assemble_constapp inst dss [] | assemble_idict (DictVar (supers, (v, (n, _)))) = fold_rev (fn super => assemble_constapp super [] o single) supers (nbe_dict v n); fun assemble_iterm constapp = let fun of_iterm match_cont t = let val (t', ts) = Code_Thingol.unfold_app t in of_iapp match_cont t' (fold_rev (cons o of_iterm NONE) ts []) end and of_iapp match_cont (IConst (c, ((_, dss), _))) ts = constapp c dss ts | of_iapp match_cont (IVar v) ts = nbe_apps (nbe_bound v) ts | of_iapp match_cont ((v, _) `|-> t) ts = nbe_apps (nbe_abss 1 (ml_abs (ml_list [nbe_bound v]) (of_iterm NONE t))) ts | of_iapp match_cont (ICase (((t, _), cs), t0)) ts = nbe_apps (ml_cases (of_iterm NONE t) (map (fn (p, t) => (of_iterm NONE p, of_iterm match_cont t)) cs @ [("_", case match_cont of SOME s => s | NONE => of_iterm NONE t0)])) ts in of_iterm end; fun subst_nonlin_vars args = let val vs = (fold o Code_Thingol.fold_varnames) (fn v => AList.map_default (op =) (v, 0) (curry (op +) 1)) args []; val names = Name.make_context (map fst vs); fun declare v k ctxt = let val vs = Name.invents ctxt v k in (vs, fold Name.declare vs ctxt) end; val (vs_renames, _) = fold_map (fn (v, k) => if k > 1 then declare v (k - 1) #>> (fn vs => (v, vs)) else pair (v, [])) vs names; val samepairs = maps (fn (v, vs) => map (pair v) vs) vs_renames; fun subst_vars (t as IConst _) samepairs = (t, samepairs) | subst_vars (t as IVar v) samepairs = (case AList.lookup (op =) samepairs v of SOME v' => (IVar v', AList.delete (op =) v samepairs) | NONE => (t, samepairs)) | subst_vars (t1 `$ t2) samepairs = samepairs |> subst_vars t1 ||>> subst_vars t2 |>> (op `$) | subst_vars (ICase (_, t)) samepairs = subst_vars t samepairs; val (args', _) = fold_map subst_vars args samepairs; in (samepairs, args') end; fun assemble_eqn c dicts default_args (i, (args, rhs)) = let val is_eval = (c = ""); val default_rhs = nbe_apps_local (i+1) c (dicts @ default_args); val match_cont = if is_eval then NONE else SOME default_rhs; val assemble_arg = assemble_iterm (fn c => fn _ => fn ts => nbe_apps_constr idx_of c ts) NONE; val assemble_rhs = assemble_iterm assemble_constapp match_cont ; val (samepairs, args') = subst_nonlin_vars args; val s_args = map assemble_arg args'; val s_rhs = if null samepairs then assemble_rhs rhs else ml_if (ml_and (map (uncurry nbe_same) samepairs)) (assemble_rhs rhs) default_rhs; val eqns = if is_eval then [([ml_list (rev (dicts @ s_args))], s_rhs)] else [([ml_list (rev (dicts @ map2 ml_as default_args s_args))], s_rhs), ([ml_list (rev (dicts @ default_args))], default_rhs)] in (nbe_fun i c, eqns) end; fun assemble_eqns (c, (num_args, (dicts, eqns))) = let val default_args = map nbe_default (Name.invent_list [] "a" (num_args - length dicts)); val eqns' = map_index (assemble_eqn c dicts default_args) eqns @ (if c = "" then [] else [(nbe_fun (length eqns) c, [([ml_list (rev (dicts @ default_args))], nbe_apps_constr idx_of c (dicts @ default_args))])]); in (eqns', nbe_abss num_args (nbe_fun 0 c)) end; val (fun_vars, fun_vals) = map_split assemble_eqns eqnss'; val deps_vars = ml_list (map (nbe_fun 0) deps); in ml_abs deps_vars (ml_Let (ml_fundefs (flat fun_vars)) (ml_list fun_vals)) end;(* code compilation *)fun compile_eqnss _ gr raw_deps [] = [] | compile_eqnss ctxt gr raw_deps eqnss = let val (deps, deps_vals) = split_list (map_filter (fn dep => Option.map (fn univ => (dep, univ)) (fst ((Graph.get_node gr dep)))) raw_deps); val idx_of = raw_deps |> map (fn dep => (dep, snd (Graph.get_node gr dep))) |> AList.lookup (op =) |> (fn f => the o f); val s = assemble_eqnss idx_of deps eqnss; val cs = map fst eqnss; in s |> tracing (fn s => "\n--- code to be evaluated:\n" ^ s) |> ML_Context.evaluate ctxt (!trace) univs_cookie |> (fn f => f deps_vals) |> (fn univs => cs ~~ univs) end;(* preparing function equations *)fun eqns_of_stmt (_, Code_Thingol.Fun (_, (_, []))) = [] | eqns_of_stmt (const, Code_Thingol.Fun (_, ((vs, _), eqns))) = [(const, (vs, map fst eqns))] | eqns_of_stmt (_, Code_Thingol.Datatypecons _) = [] | eqns_of_stmt (_, Code_Thingol.Datatype _) = [] | eqns_of_stmt (class, Code_Thingol.Class (_, (v, (superclasses, classops)))) = let val names = map snd superclasses @ map fst classops; val params = Name.invent_list [] "d" (length names); fun mk (k, name) = (name, ([(v, [])], [([IConst (class, (([], []), [])) `$$ map IVar params], IVar (nth params k))])); in map_index mk names end | eqns_of_stmt (_, Code_Thingol.Classrel _) = [] | eqns_of_stmt (_, Code_Thingol.Classparam _) = [] | eqns_of_stmt (inst, Code_Thingol.Classinst ((class, (_, arities)), (superinsts, instops))) = [(inst, (arities, [([], IConst (class, (([], []), [])) `$$ map (fn (_, (_, (inst, dicts))) => IConst (inst, (([], dicts), []))) superinsts @ map (IConst o snd o fst) instops)]))];fun compile_stmts ctxt stmts_deps = let val names = map (fst o fst) stmts_deps; val names_deps = map (fn ((name, _), deps) => (name, deps)) stmts_deps; val eqnss = maps (eqns_of_stmt o fst) stmts_deps; val refl_deps = names_deps |> maps snd |> distinct (op =) |> fold (insert (op =)) names; fun new_node name (gr, (maxidx, idx_tab)) = if can (Graph.get_node gr) name then (gr, (maxidx, idx_tab)) else (Graph.new_node (name, (NONE, maxidx)) gr, (maxidx + 1, Inttab.update_new (maxidx, name) idx_tab)); fun compile gr = eqnss |> compile_eqnss ctxt gr refl_deps |> rpair gr; in fold new_node refl_deps #> apfst (fold (fn (name, deps) => fold (curry Graph.add_edge name) deps) names_deps #> compile #-> fold (fn (name, univ) => (Graph.map_node name o apfst) (K (SOME univ)))) end;fun ensure_stmts ctxt naming program = let fun add_stmts names (gr, (maxidx, idx_tab)) = if exists ((can o Graph.get_node) gr) names then (gr, (maxidx, idx_tab)) else (gr, (maxidx, idx_tab)) |> compile_stmts ctxt (map (fn name => ((name, Graph.get_node program name), Graph.imm_succs program name)) names); in fold_rev add_stmts (Graph.strong_conn program) #> pair naming end;(** evaluation **)(* term evaluation *)fun eval_term ctxt gr deps (vs : (string * sort) list, t) = let val dict_frees = maps (fn (v, sort) => map_index (curry DFree v o fst) sort) vs; in ("", (vs, [([], t)])) |> singleton (compile_eqnss ctxt gr deps) |> snd |> (fn t => apps t (rev dict_frees)) end;(* reification *)fun typ_of_itype program vs (ityco `%% itys) = let val Code_Thingol.Datatype (tyco, _) = Graph.get_node program ityco; in Type (tyco, map (typ_of_itype program vs) itys) end | typ_of_itype program vs (ITyVar v) = let val sort = (the o AList.lookup (op =) vs) v; in TFree ("'" ^ v, sort) end;fun term_of_univ thy program idx_tab t = let fun take_until f [] = [] | take_until f (x::xs) = if f x then [] else x :: take_until f xs; fun is_dict (Const (idx, _)) = (case (Graph.get_node program o the o Inttab.lookup idx_tab) idx of Code_Thingol.Class _ => true | Code_Thingol.Classrel _ => true | Code_Thingol.Classinst _ => true | _ => false) | is_dict (DFree _) = true | is_dict _ = false; fun const_of_idx idx = (case (Graph.get_node program o the o Inttab.lookup idx_tab) idx of Code_Thingol.Fun (c, _) => c | Code_Thingol.Datatypecons (c, _) => c | Code_Thingol.Classparam (c, _) => c); fun of_apps bounds (t, ts) = fold_map (of_univ bounds) ts #>> (fn ts' => list_comb (t, rev ts')) and of_univ bounds (Const (idx, ts)) typidx = let val ts' = take_until is_dict ts; val c = const_of_idx idx; val (_, T) = Code.default_typscheme thy c; val T' = map_type_tfree (fn (v, _) => TypeInfer.param typidx (v, [])) T; val typidx' = typidx + 1; in of_apps bounds (Term.Const (c, T'), ts') typidx' end | of_univ bounds (BVar (n, ts)) typidx = of_apps bounds (Bound (bounds - n - 1), ts) typidx | of_univ bounds (t as Abs _) typidx = typidx |> of_univ (bounds + 1) (apps t [BVar (bounds, [])]) |-> (fn t' => pair (Term.Abs ("u", dummyT, t'))) in of_univ 0 t 0 |> fst end;(* function store *)structure Nbe_Functions = CodeDataFun( type T = Code_Thingol.naming * ((Univ option * int) Graph.T * (int * string Inttab.table)); val empty = (Code_Thingol.empty_naming, (Graph.empty, (0, Inttab.empty))); fun purge thy cs (naming, (gr, (maxidx, idx_tab))) = let val names_delete = cs |> map_filter (Code_Thingol.lookup_const naming) |> filter (can (Graph.get_node gr)) |> Graph.all_preds gr; val gr' = Graph.del_nodes names_delete gr; in (naming, (gr', (maxidx, idx_tab))) end;);(* compilation, evaluation and reification *)fun compile_eval thy naming program vs_t deps = let val ctxt = ProofContext.init thy; val (_, (gr, (_, idx_tab))) = Nbe_Functions.change thy (ensure_stmts ctxt naming program o snd); in vs_t |> eval_term ctxt gr deps |> term_of_univ thy program idx_tab end;(* evaluation with type reconstruction *)fun normalize thy naming program ((vs0, (vs, ty)), t) deps = let fun subst_const f = map_aterms (fn t as Term.Const (c, ty) => Term.Const (f c, ty) | t => t); val resubst_triv_consts = subst_const (Code.resubst_alias thy); val ty' = typ_of_itype program vs0 ty; fun type_infer t = singleton (TypeInfer.infer_types (Syntax.pp_global thy) (Sign.tsig_of thy) I (try (Type.strip_sorts o Sign.the_const_type thy)) (K NONE) Name.context 0) (TypeInfer.constrain ty' t); fun check_tvars t = if null (Term.add_tvars t []) then t else error ("Illegal schematic type variables in normalized term: " ^ setmp show_types true (Syntax.string_of_term_global thy) t); val string_of_term = setmp show_types true (Syntax.string_of_term_global thy); in compile_eval thy naming program (vs, t) deps |> tracing (fn t => "Normalized:\n" ^ string_of_term t) |> resubst_triv_consts |> type_infer |> tracing (fn t => "Types inferred:\n" ^ string_of_term t) |> check_tvars |> tracing (fn t => "---\n") end;(* evaluation oracle *)fun add_triv_classes thy = curry (Sorts.inter_sort (Sign.classes_of thy)) (Code.triv_classes thy);fun mk_equals thy lhs raw_rhs = let val ty = Thm.typ_of (Thm.ctyp_of_term lhs); val eq = Thm.cterm_of thy (Term.Const ("==", ty --> ty --> propT)); val rhs = Thm.cterm_of thy raw_rhs; in Thm.mk_binop eq lhs rhs end;val (_, raw_norm_oracle) = Context.>>> (Context.map_theory_result (Thm.add_oracle (Binding.name "norm", fn (thy, naming, program, vsp_ty_t, deps, ct) => mk_equals thy ct (normalize thy naming program vsp_ty_t deps))));fun norm_oracle thy naming program vsp_ty_t deps ct = raw_norm_oracle (thy, naming, program, vsp_ty_t, deps, ct);fun no_frees_conv conv ct = let val frees = Thm.add_cterm_frees ct []; fun apply_beta free thm = Thm.combination thm (Thm.reflexive free) |> Conv.fconv_rule (Conv.arg_conv (Conv.try_conv (Thm.beta_conversion false))) |> Conv.fconv_rule (Conv.arg1_conv (Thm.beta_conversion false)); in ct |> fold_rev Thm.cabs frees |> conv |> fold apply_beta frees end;fun no_frees_rew rew t = let val frees = map Free (Term.add_frees t []); in t |> fold_rev lambda frees |> rew |> (fn t' => Term.betapplys (t', frees)) end;val norm_conv = no_frees_conv (fn ct => let val thy = Thm.theory_of_cterm ct; in Code_Thingol.eval_conv thy (add_triv_classes thy) (norm_oracle thy) ct end);fun norm thy = no_frees_rew (Code_Thingol.eval thy (add_triv_classes thy) I (normalize thy));(* evaluation command *)fun norm_print_term ctxt modes t = let val thy = ProofContext.theory_of ctxt; val t' = norm thy t; val ty' = Term.type_of t'; val ctxt' = Variable.auto_fixes t ctxt; val p = PrintMode.with_modes modes (fn () => Pretty.block [Pretty.quote (Syntax.pretty_term ctxt' t'), Pretty.fbrk, Pretty.str "::", Pretty.brk 1, Pretty.quote (Syntax.pretty_typ ctxt' ty')]) (); in Pretty.writeln p end;(** Isar setup **)fun norm_print_term_cmd (modes, s) state = let val ctxt = Toplevel.context_of state in norm_print_term ctxt modes (Syntax.read_term ctxt s) end;val setup = Value.add_evaluator ("nbe", norm o ProofContext.theory_of);local structure P = OuterParse and K = OuterKeyword inval opt_modes = Scan.optional (P.$$$ "(" |-- P.!!! (Scan.repeat1 P.xname --| P.$$$ ")")) [];val _ = OuterSyntax.improper_command "normal_form" "normalize term by evaluation" K.diag (opt_modes -- P.term >> (Toplevel.keep o norm_print_term_cmd));end;end;