(* Title: HOLCF/Tools/fixrec.ML
Author: Amber Telfer and Brian Huffman
Recursive function definition package for HOLCF.
*)
signature FIXREC =
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 fixrec_simp_add: attribute
val fixrec_simp_del: attribute
val fixrec_simp_tac: Proof.context -> int -> tactic
val setup: theory -> theory
end;
structure Fixrec :> FIXREC =
struct
val def_cont_fix_eq = @{thm def_cont_fix_eq};
val def_cont_fix_ind = @{thm def_cont_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 cfun}, [T, U]);
infixr 6 ->>; val (op ->>) = cfunT;
fun dest_cfunT (Type(@{type_name cfun}, [T, U])) = (T, U)
| dest_cfunT T = raise TYPE ("dest_cfunT", [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 [] = HOLogic.unitT
| tupleT [T] = T
| tupleT (T :: Ts) = HOLogic.mk_prodT (T, tupleT Ts);
local
fun binder_cfun (Type(@{type_name cfun},[T, U])) = T :: binder_cfun U
| binder_cfun (Type(@{type_name "fun"},[T, U])) = T :: binder_cfun U
| binder_cfun _ = [];
fun body_cfun (Type(@{type_name cfun},[T, U])) = body_cfun U
| body_cfun (Type(@{type_name "fun"},[T, U])) = body_cfun U
| body_cfun T = T;
fun strip_cfun T : typ list * typ =
(binder_cfun T, body_cfun T);
fun cfunsT (Ts, U) = List.foldr cfunT U Ts;
in
fun matchT (T, U) =
body_cfun T ->> cfunsT (binder_cfun T, U) ->> U;
end
(*************************************************************************)
(***************************** 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_cabs t =
let val T = Term.fastype_of t
in cabs_const (Term.domain_type T, Term.range_type T) $ t end
fun mk_capply (t, u) =
let val (S, T) =
case Term.fastype_of t of
Type(@{type_name cfun}, [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;
(* 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);
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;
fun mk_cont t =
let val T = Term.fastype_of t
in Const(@{const_name cont}, T --> HOLogic.boolT) $ t end;
val mk_fst = HOLogic.mk_fst
val mk_snd = HOLogic.mk_snd
(* builds the expression (v1,v2,..,vn) *)
fun mk_tuple [] = HOLogic.unit
| mk_tuple (t::[]) = t
| mk_tuple (t::ts) = HOLogic.mk_prod (t, mk_tuple ts);
(* builds the expression (%(v1,v2,..,vn). rhs) *)
fun lambda_tuple [] rhs = Term.lambda (Free("unit", HOLogic.unitT)) rhs
| lambda_tuple (v::[]) rhs = Term.lambda v rhs
| lambda_tuple (v::vs) rhs =
HOLogic.mk_split (Term.lambda v (lambda_tuple vs rhs));
(*************************************************************************)
(************* fixed-point definitions and unfolding theorems ************)
(*************************************************************************)
structure FixrecUnfoldData = Generic_Data
(
type T = thm Symtab.table;
val empty = Symtab.empty;
val extend = I;
fun merge data : T = Symtab.merge (K true) data;
);
local
fun name_of (Const (n, T)) = n
| name_of (Free (n, T)) = n
| name_of _ = fixrec_err "name_of"
val lhs_name =
name_of o fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of;
in
val add_unfold : attribute =
Thm.declaration_attribute
(fn th => FixrecUnfoldData.map (Symtab.insert (K true) (lhs_name th, th)));
end
fun add_fixdefs
(fixes : ((binding * typ) * mixfix) list)
(spec : (Attrib.binding * term) list)
(lthy : local_theory) =
let
val thy = ProofContext.theory_of lthy;
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 functional = lambda_tuple lhss (mk_tuple rhss);
val fixpoint = mk_fix (mk_cabs functional);
val cont_thm =
Goal.prove lthy [] [] (mk_trp (mk_cont functional))
(K (simp_tac (simpset_of lthy) 1));
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_fst r) :: defs ls (mk_snd r);
val fixdefs = defs lhss fixpoint;
val (fixdef_thms : (term * (string * thm)) list, lthy') = lthy
|> fold_map Local_Theory.define (map (apfst fst) fixes ~~ fixdefs);
fun pair_equalI (thm1, thm2) = @{thm Pair_equalI} OF [thm1, thm2];
val tuple_fixdef_thm = foldr1 pair_equalI (map (snd o snd) fixdef_thms);
val P = Var (("P", 0), map Term.fastype_of lhss ---> HOLogic.boolT);
val predicate = lambda_tuple lhss (list_comb (P, lhss));
val tuple_induct_thm = (def_cont_fix_ind OF [tuple_fixdef_thm, cont_thm])
|> Drule.instantiate' [] [SOME (Thm.cterm_of thy predicate)]
|> LocalDefs.unfold lthy @{thms split_paired_all split_conv split_strict};
val tuple_unfold_thm = (def_cont_fix_eq OF [tuple_fixdef_thm, cont_thm])
|> LocalDefs.unfold lthy' @{thms split_conv};
fun unfolds [] thm = []
| unfolds (n::[]) thm = [(n, thm)]
| unfolds (n::ns) thm = let
val thmL = thm RS @{thm Pair_eqD1};
val thmR = thm RS @{thm Pair_eqD2};
in (n, thmL) :: unfolds ns thmR end;
val unfold_thms = unfolds names tuple_unfold_thm;
val induct_note : Attrib.binding * Thm.thm list =
let
val thm_name = Binding.name (all_names ^ "_induct");
in
((thm_name, []), [tuple_induct_thm])
end;
fun unfold_note (name, thm) : Attrib.binding * Thm.thm list =
let
val thm_name = Binding.name (name ^ "_unfold");
val src = Attrib.internal (K add_unfold);
in
((thm_name, [src]), [thm])
end;
val (thmss, lthy'') = lthy'
|> fold_map Local_Theory.note (induct_note :: map unfold_note unfold_thms);
in
(lthy'', names, fixdef_thms, map snd unfold_thms)
end;
(*************************************************************************)
(*********** monadic notation and pattern matching compilation ***********)
(*************************************************************************)
structure FixrecMatchData = Theory_Data
(
type T = string Symtab.table;
val empty = Symtab.empty;
val extend = I;
fun merge data = Symtab.merge (K true) data;
);
(* 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
| f$(v as Free(n,T)) =>
pre_build match_name f rhs (v::vs) taken
| f$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 cfun},[_,T])) (x::xs) = result_type T xs
| result_type (Type (@{type_name "fun"},[_,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, fastype_of rhs));
val k = big_lambdas vs rhs;
in
(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 **********************)
(*************************************************************************)
structure FixrecSimpData = Generic_Data
(
type T = simpset;
val empty =
HOL_basic_ss
addsimps simp_thms
addsimps [@{thm beta_cfun}]
addsimprocs [@{simproc cont_proc}];
val extend = I;
val merge = merge_ss;
);
fun fixrec_simp_tac ctxt =
let
val tab = FixrecUnfoldData.get (Context.Proof ctxt);
val ss = FixrecSimpData.get (Context.Proof ctxt);
fun concl t =
if Logic.is_all t then concl (snd (Logic.dest_all t))
else HOLogic.dest_Trueprop (Logic.strip_imp_concl t);
fun tac (t, i) =
let
val Const (c, T) = chead_of (fst (HOLogic.dest_eq (concl t)));
val unfold_thm = the (Symtab.lookup tab c);
val rule = unfold_thm RS @{thm ssubst_lhs};
in
CHANGED (rtac rule i THEN asm_simp_tac ss i)
end
in
SUBGOAL (fn ti => the_default no_tac (try tac ti))
end;
val fixrec_simp_add : attribute =
Thm.declaration_attribute
(fn th => FixrecSimpData.map (fn ss => ss addsimps [th]));
val fixrec_simp_del : attribute =
Thm.declaration_attribute
(fn th => FixrecSimpData.map (fn ss => ss delsimps [th]));
(* proves a block of pattern matching equations as theorems, using unfold *)
fun make_simps ctxt (unfold_thm, eqns : (Attrib.binding * term) list) =
let
val tacs =
[rtac (unfold_thm RS @{thm ssubst_lhs}) 1,
asm_simp_tac (simpset_of ctxt) 1];
fun prove_term t = Goal.prove ctxt [] [] 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.ML *)
fun gen_fixrec
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])) (flat simps);
val (_, lthy'') = lthy'
|> fold_map Local_Theory.note (simps1 @ simps2);
in
lthy''
end
else lthy'
end;
in
val add_fixrec = gen_fixrec Specification.check_spec;
val add_fixrec_cmd = gen_fixrec 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 (global_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 =
Attrib.setup @{binding fixrec_simp}
(Attrib.add_del fixrec_simp_add fixrec_simp_del)
"declaration of fixrec simp rule"
#> Method.setup @{binding fixrec_simp}
(Scan.succeed (SIMPLE_METHOD' o fixrec_simp_tac))
"pattern prover for fixrec constants";
end;