src/HOLCF/Tools/fixrec.ML

+(*  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 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 "->"}, [T, U]);
+
+infixr 6 ->>; val (op ->>) = cfunT;
+
+fun cfunsT (Ts, U) = foldr cfunT U Ts;
+
+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 [] = HOLogic.unitT
+  | tupleT [T] = T
+  | tupleT (T :: Ts) = HOLogic.mk_prodT (T, tupleT Ts);
+
+fun matchT (T, U) =
+  body_cfun T ->> cfunsT (binder_cfun T, U) ->> U;
+
+
+(*************************************************************************)
+(***************************** 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 "->"}, [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 ************)
+(*************************************************************************)
+
+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 (local_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 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 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^"_unfold", thm)]
+      | unfolds (n::ns) thm = let
+          val thmL = thm RS @{thm Pair_eqD1};
+          val thmR = thm RS @{thm Pair_eqD2};
+        in (n^"_unfold", thmL) :: unfolds ns thmR end;
+    val unfold_thms = unfolds names tuple_unfold_thm;
+    fun mk_note (n, thm) = ((Binding.name n, []), [thm]);
+    val (thmss, lthy'') = lthy'
+      |> fold_map (LocalTheory.note Thm.generatedK o mk_note)
+        ((all_names ^ "_induct", tuple_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, 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 **********************)
+(*************************************************************************)
+
+(* 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.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.generatedK) (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;