(*  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

  in Const(@{const_name Fixrec.run}, mT ->> T) ` 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" ^

          "are configured for all non-HOLCF constants.\n" ^

          "The error occurred for the goal statement:\n" ^

          Syntax.string_of_term lthy prop);

        fun check th = case Thm.nprems_of th of 0 => all_tac th | n => err ();

        val tac = simp_tac (simpset_of lthy) 1 THEN check;

      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 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, matcherT (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_matchT U) (mk_run msum)

  end;

(* this is the pattern-matching compiler function *)

fun compile_pats match_name eqs =

  let

    val ((names, arities), mats) =

      apfst ListPair.unzip (ListPair.unzip (map (match_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 **********************)

(*************************************************************************)

(* tactic to perform eta-contraction on subgoal *)

fun eta_tac (ctxt : Proof.context) : int -> tactic =

  EqSubst.eqsubst_tac ctxt [0] @{thms refl};

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 ctxt 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 ctxt 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_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.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 = 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 _ = 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 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 ********************************)

(*************************************************************************)

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;