(* 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_tac: Proof.context -> int -> tactic
val setup: theory -> theory
end;
structure Fixrec :> FIXREC =
struct
open HOLCF_Library;
infixr 6 ->>;
infix -->>;
infix 9 `;
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 ****************************)
(*************************************************************************)
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);
in
fun matcherT (T, U) =
body_cfun T ->> (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;
infix 0 ==; val (op ==) = Logic.mk_equals;
infix 1 ===; val (op ===) = HOLogic.mk_eq;
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_matchT mT
val run = Const(@{const_name Fixrec.run}, mT ->> T)
in
case t of
Const(@{const_name Rep_CFun}, _) $
Const(@{const_name Fixrec.succeed}, _) $ u => u
| _ => run ` t
end;
(*************************************************************************)
(************* 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 t = raise TERM ("Fixrec.add_unfold: lhs not a constant", [t]);
val lhs_name =
name_of o head_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 =
let
val prop = mk_trp (mk_cont functional);
fun err _ = error (
"Continuity proof failed; please check that cont2cont rules\n" ^
"or simp rules are configured for all non-HOLCF constants.\n" ^
"The error occurred for the goal statement:\n" ^
Syntax.string_of_term lthy prop);
val rules = Cont2ContData.get lthy;
val fast_tac = SOLVED' (REPEAT_ALL_NEW (match_tac rules));
val slow_tac = SOLVED' (simp_tac (simpset_of lthy));
val tac = fast_tac 1 ORELSE slow_tac 1 ORELSE err;
in
Goal.prove lthy [] [] prop (K tac)
end;
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)]
|> Local_Defs.unfold lthy @{thms split_paired_all split_conv split_strict};
val tuple_unfold_thm = (def_cont_fix_eq OF [tuple_fixdef_thm, cont_thm])
|> Local_Defs.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.qualify true all_names (Binding.name "induct");
in
((thm_name, []), [tuple_induct_thm])
end;
fun unfold_note (name, thm) : Attrib.binding * Thm.thm list =
let
val thm_name = Binding.qualify true name (Binding.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 pattern *)
(* returns (rhs, free variable, used varnames) *)
fun compile_pat match_name pat rhs taken =
let
fun comp_pat p rhs taken =
if is_Free p then (rhs, p, taken)
else comp_con (fastype_of p) p rhs [] taken
(* compiles a monadic term for a constructor pattern *)
and comp_con T p rhs vs taken =
case p of
Const(@{const_name Rep_CFun},_) $ f $ x =>
let val (rhs', v, taken') = comp_pat x rhs taken
in comp_con T f rhs' (v::vs) taken' end
| f $ x =>
let val (rhs', v, taken') = comp_pat x rhs taken
in comp_con T f rhs' (v::vs) taken' end
| Const (c, cT) =>
let
val n = Name.variant taken "v"
val v = Free(n, T)
val m = Const(match_name c, matcherT (cT, fastype_of rhs))
val k = big_lambdas vs rhs
in
(m`v`k, v, n::taken)
end
| _ => raise TERM ("fixrec: invalid pattern ", [p])
in
comp_pat pat rhs taken
end;
(* builds a monadic term for matching a function definition pattern *)
(* returns (name, arity, matcher) *)
fun compile_lhs match_name pat rhs vs taken =
case pat of
Const(@{const_name Rep_CFun}, _) $ f $ x =>
let val (rhs', v, taken') = compile_pat match_name x rhs taken;
in compile_lhs 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 compile_eq match_name eq =
let
val (lhs,rhs) = dest_eqs (Logic.strip_imp_concl (strip_alls eq));
in
compile_lhs match_name lhs (mk_succeed 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_matchT U) (mk_run msum)
end;
(* this is the pattern-matching compiler function *)
fun compile_eqs match_name eqs =
let
val ((names, arities), mats) =
apfst ListPair.unzip (ListPair.unzip (map (compile_eq match_name) eqs));
val cname =
case distinct (op =) names of
[n] => n
| _ => fixrec_err "all equations in block must define the same function";
val arity =
case distinct (op =) arities of
[a] => a
| _ => 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 **********************)
(*************************************************************************)
fun eta_tac i = CONVERSION Thm.eta_conversion i;
fun fixrec_simp_tac ctxt =
let
val tab = FixrecUnfoldData.get (Context.Proof ctxt);
val ss = Simplifier.simpset_of 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 (c, T) =
(dest_Const o head_of o chead_of o fst o HOLogic.dest_eq o concl) t;
val unfold_thm = the (Symtab.lookup tab c);
val rule = unfold_thm RS @{thm ssubst_lhs};
in
CHANGED (rtac rule i THEN eta_tac i THEN asm_simp_tac ss i)
end
in
SUBGOAL (fn ti => the_default no_tac (try tac ti))
end;
(* proves a block of pattern matching equations as theorems, using unfold *)
fun make_simps ctxt (unfold_thm, eqns : (Attrib.binding * term) list) =
let
val ss = Simplifier.simpset_of ctxt;
val rule = unfold_thm RS @{thm ssubst_lhs};
val tac = rtac rule 1 THEN eta_tac 1 THEN asm_simp_tac ss 1;
fun prove_term t = Goal.prove ctxt [] [] t (K tac);
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_eqs 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.qualify true n (Binding.name "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 = Global_Theory.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 _ = legacy_feature "Old 'fixpat' command -- use \"fixrec_simp\" method instead";
val atts = map (prep_attrib thy) srcs;
val ts = map (prep_term thy) strings;
val simps = map (fix_pat thy) ts;
in
(snd o Global_Theory.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 ********************************)
(*************************************************************************)
val _ =
Outer_Syntax.local_theory "fixrec" "define recursive functions (HOLCF)" Keyword.thy_decl
((Parse.opt_keyword "permissive" >> not) -- Parse.fixes -- Parse_Spec.where_alt_specs
>> (fn ((strict, fixes), specs) => add_fixrec_cmd strict fixes specs));
val _ =
Outer_Syntax.command "fixpat" "define rewrites for fixrec functions" Keyword.thy_decl
(Parse_Spec.specs >> (Toplevel.theory o add_fixpat_cmd));
val setup =
Method.setup @{binding fixrec_simp}
(Scan.succeed (SIMPLE_METHOD' o fixrec_simp_tac))
"pattern prover for fixrec constants";
end;