(* Title: HOLCF/Tools/fixrec_package.ML Author: Amber Telfer and Brian Huffman Recursive function definition package for HOLCF. *) signature FIXREC_PACKAGE = sig val add_fixrec: bool -> (binding * typ option * mixfix) list -> (Attrib.binding * term) list -> local_theory -> local_theory val add_fixrec_cmd: bool -> (binding * string option * mixfix) list -> (Attrib.binding * string) list -> local_theory -> local_theory val add_fixpat: Thm.binding * term list -> theory -> theory val add_fixpat_cmd: Attrib.binding * string list -> theory -> theory val add_matchers: (string * string) list -> theory -> theory val setup: theory -> theory end; structure FixrecPackage: FIXREC_PACKAGE = struct val fix_eq2 = @{thm fix_eq2}; val def_fix_ind = @{thm def_fix_ind}; fun fixrec_err s = error ("fixrec definition error:\n" ^ s); fun fixrec_eq_err thy s eq = fixrec_err (s ^ "\nin\n" ^ quote (Syntax.string_of_term_global thy eq)); (*************************************************************************) (***************************** building types ****************************) (*************************************************************************) (* ->> is taken from holcf_logic.ML *) fun cfunT (T, U) = Type(@{type_name "->"}, [T, U]); infixr 6 ->>; val (op ->>) = cfunT; fun dest_cfunT (Type(@{type_name "->"}, [T, U])) = (T, U) | dest_cfunT T = raise TYPE ("dest_cfunT", [T], []); fun binder_cfun (Type(@{type_name "->"},[T, U])) = T :: binder_cfun U | binder_cfun _ = []; fun body_cfun (Type(@{type_name "->"},[T, U])) = body_cfun U | body_cfun T = T; fun strip_cfun T : typ list * typ = (binder_cfun T, body_cfun T); fun maybeT T = Type(@{type_name "maybe"}, [T]); fun dest_maybeT (Type(@{type_name "maybe"}, [T])) = T | dest_maybeT T = raise TYPE ("dest_maybeT", [T], []); fun tupleT [] = @{typ "unit"} | tupleT [T] = T | tupleT (T :: Ts) = HOLogic.mk_prodT (T, tupleT Ts); fun matchT T = body_cfun T ->> maybeT (tupleT (binder_cfun T)); (*************************************************************************) (***************************** building terms ****************************) (*************************************************************************) val mk_trp = HOLogic.mk_Trueprop; (* splits a cterm into the right and lefthand sides of equality *) fun dest_eqs t = HOLogic.dest_eq (HOLogic.dest_Trueprop t); (* similar to Thm.head_of, but for continuous application *) fun chead_of (Const(@{const_name Rep_CFun},_)$f$t) = chead_of f | chead_of u = u; fun capply_const (S, T) = Const(@{const_name Rep_CFun}, (S ->> T) --> (S --> T)); fun cabs_const (S, T) = Const(@{const_name Abs_CFun}, (S --> T) --> (S ->> T)); fun mk_capply (t, u) = let val (S, T) = case Term.fastype_of t of Type(@{type_name "->"}, [S, T]) => (S, T) | _ => raise TERM ("mk_capply " ^ ML_Syntax.print_list ML_Syntax.print_term [t, u], [t, u]); in capply_const (S, T) $ t $ u end; infix 0 ==; val (op ==) = Logic.mk_equals; infix 1 ===; val (op ===) = HOLogic.mk_eq; infix 9 ` ; val (op `) = mk_capply; fun mk_cpair (t, u) = let val T = Term.fastype_of t val U = Term.fastype_of u val cpairT = T ->> U ->> HOLogic.mk_prodT (T, U) in Const(@{const_name cpair}, cpairT) ` t ` u end; fun mk_cfst t = let val T = Term.fastype_of t; val (U, _) = HOLogic.dest_prodT T; in Const(@{const_name cfst}, T ->> U) ` t end; fun mk_csnd t = let val T = Term.fastype_of t; val (_, U) = HOLogic.dest_prodT T; in Const(@{const_name csnd}, T ->> U) ` t end; fun mk_csplit t = let val (S, TU) = dest_cfunT (Term.fastype_of t); val (T, U) = dest_cfunT TU; val csplitT = (S ->> T ->> U) ->> HOLogic.mk_prodT (S, T) ->> U; in Const(@{const_name csplit}, csplitT) ` t end; (* builds the expression (LAM v. rhs) *) fun big_lambda v rhs = cabs_const (Term.fastype_of v, Term.fastype_of rhs) $ Term.lambda v rhs; (* builds the expression (LAM v1 v2 .. vn. rhs) *) fun big_lambdas [] rhs = rhs | big_lambdas (v::vs) rhs = big_lambda v (big_lambdas vs rhs); (* builds the expression (LAM <v1,v2,..,vn>. rhs) *) fun lambda_ctuple [] rhs = big_lambda (Free("unit", HOLogic.unitT)) rhs | lambda_ctuple (v::[]) rhs = big_lambda v rhs | lambda_ctuple (v::vs) rhs = mk_csplit (big_lambda v (lambda_ctuple vs rhs)); (* builds the expression <v1,v2,..,vn> *) fun mk_ctuple [] = @{term "UU::unit"} | mk_ctuple (t::[]) = t | mk_ctuple (t::ts) = mk_cpair (t, mk_ctuple ts); fun mk_return t = let val T = Term.fastype_of t in Const(@{const_name Fixrec.return}, T ->> maybeT T) ` t end; fun mk_bind (t, u) = let val (T, mU) = dest_cfunT (Term.fastype_of u); val bindT = maybeT T ->> (T ->> mU) ->> mU; in Const(@{const_name Fixrec.bind}, bindT) ` t ` u end; fun mk_mplus (t, u) = let val mT = Term.fastype_of t in Const(@{const_name Fixrec.mplus}, mT ->> mT ->> mT) ` t ` u end; fun mk_run t = let val mT = Term.fastype_of t val T = dest_maybeT mT in Const(@{const_name Fixrec.run}, mT ->> T) ` t end; fun mk_fix t = let val (T, _) = dest_cfunT (Term.fastype_of t) in Const(@{const_name fix}, (T ->> T) ->> T) ` t end; (*************************************************************************) (************* fixed-point definitions and unfolding theorems ************) (*************************************************************************) fun add_fixdefs (fixes : ((binding * typ) * mixfix) list) (spec : (Attrib.binding * term) list) (lthy : local_theory) = let val names = map (Binding.name_of o fst o fst) fixes; val all_names = space_implode "_" names; val (lhss,rhss) = ListPair.unzip (map (dest_eqs o snd) spec); val fixpoint = mk_fix (lambda_ctuple lhss (mk_ctuple rhss)); fun one_def (l as Free(n,_)) r = let val b = Long_Name.base_name n in ((Binding.name (b^"_def"), []), r) end | one_def _ _ = fixrec_err "fixdefs: lhs not of correct form"; fun defs [] _ = [] | defs (l::[]) r = [one_def l r] | defs (l::ls) r = one_def l (mk_cfst r) :: defs ls (mk_csnd r); val fixdefs = defs lhss fixpoint; val define_all = fold_map (LocalTheory.define Thm.definitionK); val (fixdef_thms : (term * (string * thm)) list, lthy') = lthy |> define_all (map (apfst fst) fixes ~~ fixdefs); fun cpair_equalI (thm1, thm2) = @{thm cpair_equalI} OF [thm1, thm2]; val ctuple_fixdef_thm = foldr1 cpair_equalI (map (snd o snd) fixdef_thms); val ctuple_induct_thm = ctuple_fixdef_thm RS def_fix_ind; val ctuple_unfold_thm = Goal.prove lthy' [] [] (mk_trp (mk_ctuple lhss === mk_ctuple rhss)) (fn _ => EVERY [rtac (ctuple_fixdef_thm RS fix_eq2 RS trans) 1, simp_tac (local_simpset_of lthy') 1]); fun unfolds [] thm = [] | unfolds (n::[]) thm = [(n^"_unfold", thm)] | unfolds (n::ns) thm = let val thmL = thm RS @{thm cpair_eqD1}; val thmR = thm RS @{thm cpair_eqD2}; in (n^"_unfold", thmL) :: unfolds ns thmR end; val unfold_thms = unfolds names ctuple_unfold_thm; fun mk_note (n, thm) = ((Binding.name n, []), [thm]); val (thmss, lthy'') = lthy' |> fold_map (LocalTheory.note Thm.theoremK o mk_note) ((all_names ^ "_induct", ctuple_induct_thm) :: unfold_thms); in (lthy'', names, fixdef_thms, map snd unfold_thms) end; (*************************************************************************) (*********** monadic notation and pattern matching compilation ***********) (*************************************************************************) structure FixrecMatchData = TheoryDataFun ( type T = string Symtab.table; val empty = Symtab.empty; val copy = I; val extend = I; fun merge _ tabs : T = Symtab.merge (K true) tabs; ); (* associate match functions with pattern constants *) fun add_matchers ms = FixrecMatchData.map (fold Symtab.update ms); fun taken_names (t : term) : bstring list = let fun taken (Const(a,_), bs) = insert (op =) (Long_Name.base_name a) bs | taken (Free(a,_) , bs) = insert (op =) a bs | taken (f $ u , bs) = taken (f, taken (u, bs)) | taken (Abs(a,_,t), bs) = taken (t, insert (op =) a bs) | taken (_ , bs) = bs; in taken (t, []) end; (* builds a monadic term for matching a constructor pattern *) fun pre_build match_name pat rhs vs taken = case pat of Const(@{const_name Rep_CFun},_)$f$(v as Free(n,T)) => pre_build match_name f rhs (v::vs) taken | Const(@{const_name Rep_CFun},_)$f$x => let val (rhs', v, taken') = pre_build match_name x rhs [] taken; in pre_build match_name f rhs' (v::vs) taken' end | Const(c,T) => let val n = Name.variant taken "v"; fun result_type (Type(@{type_name "->"},[_,T])) (x::xs) = result_type T xs | result_type T _ = T; val v = Free(n, result_type T vs); val m = Const(match_name c, matchT T); val k = lambda_ctuple vs rhs; in (mk_bind (m`v, k), v, n::taken) end | Free(n,_) => fixrec_err ("expected constructor, found free variable " ^ quote n) | _ => fixrec_err "pre_build: invalid pattern"; (* builds a monadic term for matching a function definition pattern *) (* returns (name, arity, matcher) *) fun building match_name pat rhs vs taken = case pat of Const(@{const_name Rep_CFun}, _)$f$(v as Free(n,T)) => building match_name f rhs (v::vs) taken | Const(@{const_name Rep_CFun}, _)$f$x => let val (rhs', v, taken') = pre_build match_name x rhs [] taken; in building match_name f rhs' (v::vs) taken' end | Free(_,_) => ((pat, length vs), big_lambdas vs rhs) | Const(_,_) => ((pat, length vs), big_lambdas vs rhs) | _ => fixrec_err ("function is not declared as constant in theory: " ^ ML_Syntax.print_term pat); fun strip_alls t = if Logic.is_all t then strip_alls (snd (Logic.dest_all t)) else t; fun match_eq match_name eq = let val (lhs,rhs) = dest_eqs (Logic.strip_imp_concl (strip_alls eq)); in building match_name lhs (mk_return rhs) [] (taken_names eq) end; (* returns the sum (using +++) of the terms in ms *) (* also applies "run" to the result! *) fun fatbar arity ms = let fun LAM_Ts 0 t = ([], Term.fastype_of t) | LAM_Ts n (_ $ Abs(_,T,t)) = let val (Ts, U) = LAM_Ts (n-1) t in (T::Ts, U) end | LAM_Ts _ _ = fixrec_err "fatbar: internal error, not enough LAMs"; fun unLAM 0 t = t | unLAM n (_$Abs(_,_,t)) = unLAM (n-1) t | unLAM _ _ = fixrec_err "fatbar: internal error, not enough LAMs"; fun reLAM ([], U) t = t | reLAM (T::Ts, U) t = reLAM (Ts, T ->> U) (cabs_const(T,U)$Abs("",T,t)); val msum = foldr1 mk_mplus (map (unLAM arity) ms); val (Ts, U) = LAM_Ts arity (hd ms) in reLAM (rev Ts, dest_maybeT U) (mk_run msum) end; (* this is the pattern-matching compiler function *) fun compile_pats match_name eqs = let val (((n::names),(a::arities)),mats) = apfst ListPair.unzip (ListPair.unzip (map (match_eq match_name) eqs)); val cname = if forall (fn x => n=x) names then n else fixrec_err "all equations in block must define the same function"; val arity = if forall (fn x => a=x) arities then a else fixrec_err "all equations in block must have the same arity"; val rhs = fatbar arity mats; in mk_trp (cname === rhs) end; (*************************************************************************) (********************** Proving associated theorems **********************) (*************************************************************************) (* proves a block of pattern matching equations as theorems, using unfold *) fun make_simps lthy (unfold_thm, eqns : (Attrib.binding * term) list) = let val tacs = [rtac (unfold_thm RS @{thm ssubst_lhs}) 1, asm_simp_tac (local_simpset_of lthy) 1]; fun prove_term t = Goal.prove lthy [] [] t (K (EVERY tacs)); fun prove_eqn (bind, eqn_t) = (bind, prove_term eqn_t); in map prove_eqn eqns end; (*************************************************************************) (************************* Main fixrec function **************************) (*************************************************************************) local (* code adapted from HOL/Tools/primrec_package.ML *) fun gen_fixrec (set_group : bool) prep_spec (strict : bool) raw_fixes raw_spec (lthy : local_theory) = let val (fixes : ((binding * typ) * mixfix) list, spec : (Attrib.binding * term) list) = fst (prep_spec raw_fixes raw_spec lthy); val chead_of_spec = chead_of o fst o dest_eqs o Logic.strip_imp_concl o strip_alls o snd; fun name_of (Free (n, _)) = n | name_of t = fixrec_err ("unknown term"); val all_names = map (name_of o chead_of_spec) spec; val names = distinct (op =) all_names; fun block_of_name n = map_filter (fn (m,eq) => if m = n then SOME eq else NONE) (all_names ~~ spec); val blocks = map block_of_name names; val matcher_tab = FixrecMatchData.get (ProofContext.theory_of lthy); fun match_name c = case Symtab.lookup matcher_tab c of SOME m => m | NONE => fixrec_err ("unknown pattern constructor: " ^ c); val matches = map (compile_pats match_name) (map (map snd) blocks); val spec' = map (pair Attrib.empty_binding) matches; val (lthy', cnames, fixdef_thms, unfold_thms) = add_fixdefs fixes spec' lthy; in if strict then let (* only prove simp rules if strict = true *) val simps : (Attrib.binding * thm) list list = map (make_simps lthy') (unfold_thms ~~ blocks); fun mk_bind n : Attrib.binding = (Binding.name (n ^ "_simps"), [Attrib.internal (K Simplifier.simp_add)]); val simps1 : (Attrib.binding * thm list) list = map (fn (n,xs) => (mk_bind n, map snd xs)) (names ~~ simps); val simps2 : (Attrib.binding * thm list) list = map (apsnd (fn thm => [thm])) (List.concat simps); val (_, lthy'') = lthy' |> fold_map (LocalTheory.note Thm.theoremK) (simps1 @ simps2); in lthy'' end else lthy' end; in val add_fixrec = gen_fixrec false Specification.check_spec; val add_fixrec_cmd = gen_fixrec true Specification.read_spec; end; (* local *) (*************************************************************************) (******************************** Fixpat *********************************) (*************************************************************************) fun fix_pat thy t = let val T = fastype_of t; val eq = mk_trp (HOLogic.eq_const T $ t $ Var (("x",0),T)); val cname = case chead_of t of Const(c,_) => c | _ => fixrec_err "function is not declared as constant in theory"; val unfold_thm = PureThy.get_thm thy (cname^"_unfold"); val simp = Goal.prove_global thy [] [] eq (fn _ => EVERY [stac unfold_thm 1, simp_tac (simpset_of thy) 1]); in simp end; fun gen_add_fixpat prep_term prep_attrib ((name, srcs), strings) thy = let val atts = map (prep_attrib thy) srcs; val ts = map (prep_term thy) strings; val simps = map (fix_pat thy) ts; in (snd o PureThy.add_thmss [((name, simps), atts)]) thy end; val add_fixpat = gen_add_fixpat Sign.cert_term (K I); val add_fixpat_cmd = gen_add_fixpat Syntax.read_term_global Attrib.attribute; (*************************************************************************) (******************************** Parsers ********************************) (*************************************************************************) local structure P = OuterParse and K = OuterKeyword in val _ = OuterSyntax.local_theory "fixrec" "define recursive functions (HOLCF)" K.thy_decl ((P.opt_keyword "permissive" >> not) -- P.fixes -- SpecParse.where_alt_specs >> (fn ((strict, fixes), specs) => add_fixrec_cmd strict fixes specs)); val _ = OuterSyntax.command "fixpat" "define rewrites for fixrec functions" K.thy_decl (SpecParse.specs >> (Toplevel.theory o add_fixpat_cmd)); end; val setup = FixrecMatchData.init; end;