src/HOL/HOLCF/Tools/fixrec.ML
author huffman
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(*  Title:      HOL/HOLCF/Tools/fixrec.ML
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    Author:     Amber Telfer and Brian Huffman
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Recursive function definition package for HOLCF.
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*)
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signature FIXREC =
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sig
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  val add_fixrec: (binding * typ option * mixfix) list
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    -> (bool * (Attrib.binding * term)) list -> local_theory -> local_theory
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  val add_fixrec_cmd: (binding * string option * mixfix) list
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    -> (bool * (Attrib.binding * string)) list -> local_theory -> local_theory
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  val add_matchers: (string * string) list -> theory -> theory
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  val fixrec_simp_tac: Proof.context -> int -> tactic
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  val setup: theory -> theory
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end
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structure Fixrec : FIXREC =
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struct
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open HOLCF_Library
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infixr 6 ->>
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infix -->>
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infix 9 `
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val def_cont_fix_eq = @{thm def_cont_fix_eq}
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val def_cont_fix_ind = @{thm def_cont_fix_ind}
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fun fixrec_err s = error ("fixrec definition error:\n" ^ s)
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(*************************************************************************)
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(***************************** building types ****************************)
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(*************************************************************************)
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local
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fun binder_cfun (Type(@{type_name cfun},[T, U])) = T :: binder_cfun U
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  | binder_cfun (Type(@{type_name "fun"},[T, U])) = T :: binder_cfun U
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  | binder_cfun _   =  []
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fun body_cfun (Type(@{type_name cfun},[_, U])) = body_cfun U
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  | body_cfun (Type(@{type_name "fun"},[_, U])) = body_cfun U
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  | body_cfun T   =  T
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in
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fun matcherT (T, U) =
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  body_cfun T ->> (binder_cfun T -->> U) ->> U
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end
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(*************************************************************************)
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(***************************** building terms ****************************)
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(*************************************************************************)
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val mk_trp = HOLogic.mk_Trueprop
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(* splits a cterm into the right and lefthand sides of equality *)
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fun dest_eqs t = HOLogic.dest_eq (HOLogic.dest_Trueprop t)
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(* similar to Thm.head_of, but for continuous application *)
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fun chead_of (Const(@{const_name Rep_cfun},_)$f$_) = chead_of f
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  | chead_of u = u
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infix 1 === val (op ===) = HOLogic.mk_eq
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fun mk_mplus (t, u) =
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  let val mT = Term.fastype_of t
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  in Const(@{const_name Fixrec.mplus}, mT ->> mT ->> mT) ` t ` u end
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fun mk_run t =
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  let
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    val mT = Term.fastype_of t
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    val T = dest_matchT mT
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    val run = Const(@{const_name Fixrec.run}, mT ->> T)
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  in
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    case t of
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      Const(@{const_name Rep_cfun}, _) $
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        Const(@{const_name Fixrec.succeed}, _) $ u => u
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    | _ => run ` t
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  end
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(*************************************************************************)
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(************* fixed-point definitions and unfolding theorems ************)
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(*************************************************************************)
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structure FixrecUnfoldData = Generic_Data
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(
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  type T = thm Symtab.table
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  val empty = Symtab.empty
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  val extend = I
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  fun merge data : T = Symtab.merge (K true) data
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)
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local
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fun name_of (Const (n, _)) = n
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  | name_of (Free (n, _)) = n
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  | name_of t = raise TERM ("Fixrec.add_unfold: lhs not a constant", [t])
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val lhs_name =
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  name_of o head_of o fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of
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in
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val add_unfold : attribute =
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  Thm.declaration_attribute
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    (fn th => FixrecUnfoldData.map (Symtab.insert (K true) (lhs_name th, th)))
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end
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fun add_fixdefs
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  (fixes : ((binding * typ) * mixfix) list)
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  (spec : (Attrib.binding * term) list)
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  (lthy : local_theory) =
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  let
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    val thy = Proof_Context.theory_of lthy
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    val names = map (Binding.name_of o fst o fst) fixes
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    val all_names = space_implode "_" names
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    val (lhss, rhss) = ListPair.unzip (map (dest_eqs o snd) spec)
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    val functional = lambda_tuple lhss (mk_tuple rhss)
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    val fixpoint = mk_fix (mk_cabs functional)
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    val cont_thm =
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      let
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        val prop = mk_trp (mk_cont functional)
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        fun err _ = error (
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          "Continuity proof failed please check that cont2cont rules\n" ^
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          "or simp rules are configured for all non-HOLCF constants.\n" ^
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          "The error occurred for the goal statement:\n" ^
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          Syntax.string_of_term lthy prop)
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        val rules = Cont2ContData.get lthy
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        val fast_tac = SOLVED' (REPEAT_ALL_NEW (match_tac rules))
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        val slow_tac = SOLVED' (simp_tac (simpset_of lthy))
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        val tac = fast_tac 1 ORELSE slow_tac 1 ORELSE err
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      in
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        Goal.prove lthy [] [] prop (K tac)
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      end
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    fun one_def (Free(n,_)) r =
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          let val b = Long_Name.base_name n
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          in ((Binding.name (b^"_def"), []), r) end
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      | one_def _ _ = fixrec_err "fixdefs: lhs not of correct form"
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    fun defs [] _ = []
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      | defs (l::[]) r = [one_def l r]
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      | defs (l::ls) r = one_def l (mk_fst r) :: defs ls (mk_snd r)
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    val fixdefs = defs lhss fixpoint
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    val (fixdef_thms : (term * (string * thm)) list, lthy) = lthy
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      |> fold_map Local_Theory.define (map (apfst fst) fixes ~~ fixdefs)
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    fun pair_equalI (thm1, thm2) = @{thm Pair_equalI} OF [thm1, thm2]
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    val tuple_fixdef_thm = foldr1 pair_equalI (map (snd o snd) fixdef_thms)
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    val P = Var (("P", 0), map Term.fastype_of lhss ---> HOLogic.boolT)
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    val predicate = lambda_tuple lhss (list_comb (P, lhss))
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    val tuple_induct_thm = (def_cont_fix_ind OF [tuple_fixdef_thm, cont_thm])
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      |> Drule.instantiate' [] [SOME (Thm.cterm_of thy predicate)]
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      |> Local_Defs.unfold lthy @{thms split_paired_all split_conv split_strict}
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    val tuple_unfold_thm = (def_cont_fix_eq OF [tuple_fixdef_thm, cont_thm])
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      |> Local_Defs.unfold lthy @{thms split_conv}
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    fun unfolds [] _ = []
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      | unfolds (n::[]) thm = [(n, thm)]
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      | unfolds (n::ns) thm = let
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          val thmL = thm RS @{thm Pair_eqD1}
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          val thmR = thm RS @{thm Pair_eqD2}
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        in (n, thmL) :: unfolds ns thmR end
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    val unfold_thms = unfolds names tuple_unfold_thm
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    val induct_note : Attrib.binding * Thm.thm list =
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      let
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        val thm_name = Binding.qualify true all_names (Binding.name "induct")
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      in
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        ((thm_name, []), [tuple_induct_thm])
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      end
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    fun unfold_note (name, thm) : Attrib.binding * Thm.thm list =
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      let
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        val thm_name = Binding.qualify true name (Binding.name "unfold")
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        val src = Attrib.internal (K add_unfold)
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      in
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        ((thm_name, [src]), [thm])
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      end
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    val (_, lthy) = lthy
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      |> fold_map Local_Theory.note (induct_note :: map unfold_note unfold_thms)
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  in
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    (lthy, names, fixdef_thms, map snd unfold_thms)
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  end
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(*************************************************************************)
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(*********** monadic notation and pattern matching compilation ***********)
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(*************************************************************************)
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structure FixrecMatchData = Theory_Data
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(
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  type T = string Symtab.table
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  val empty = Symtab.empty
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  val extend = I
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  fun merge data = Symtab.merge (K true) data
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)
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(* associate match functions with pattern constants *)
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fun add_matchers ms = FixrecMatchData.map (fold Symtab.update ms)
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fun taken_names (t : term) : bstring list =
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  let
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    fun taken (Const(a,_), bs) = insert (op =) (Long_Name.base_name a) bs
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      | taken (Free(a,_) , bs) = insert (op =) a bs
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      | taken (f $ u     , bs) = taken (f, taken (u, bs))
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      | taken (Abs(a,_,t), bs) = taken (t, insert (op =) a bs)
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      | taken (_         , bs) = bs
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  in
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    taken (t, [])
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  end
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(* builds a monadic term for matching a pattern *)
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(* returns (rhs, free variable, used varnames) *)
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fun compile_pat match_name pat rhs taken =
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  let
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    fun comp_pat p rhs taken =
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      if is_Free p then (rhs, p, taken)
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      else comp_con (fastype_of p) p rhs [] taken
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    (* compiles a monadic term for a constructor pattern *)
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    and comp_con T p rhs vs taken =
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      case p of
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        Const(@{const_name Rep_cfun},_) $ f $ x =>
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          let val (rhs', v, taken') = comp_pat x rhs taken
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          in comp_con T f rhs' (v::vs) taken' end
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      | f $ x =>
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          let val (rhs', v, taken') = comp_pat x rhs taken
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          in comp_con T f rhs' (v::vs) taken' end
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      | Const (c, cT) =>
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          let
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            val n = singleton (Name.variant_list taken) "v"
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            val v = Free(n, T)
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            val m = Const(match_name c, matcherT (cT, fastype_of rhs))
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            val k = big_lambdas vs rhs
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          in
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            (m`v`k, v, n::taken)
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          end
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      | _ => raise TERM ("fixrec: invalid pattern ", [p])
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  in
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    comp_pat pat rhs taken
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  end
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(* builds a monadic term for matching a function definition pattern *)
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(* returns (constant, (vars, matcher)) *)
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fun compile_lhs match_name pat rhs vs taken =
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  case pat of
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    Const(@{const_name Rep_cfun}, _) $ f $ x =>
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      let val (rhs', v, taken') = compile_pat match_name x rhs taken
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      in compile_lhs match_name f rhs' (v::vs) taken' end
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  | Free(_,_) => (pat, (vs, rhs))
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  | Const(_,_) => (pat, (vs, rhs))
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  | _ => fixrec_err ("invalid function pattern: "
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                    ^ ML_Syntax.print_term pat)
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fun strip_alls t =
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  if Logic.is_all t then strip_alls (snd (Logic.dest_all t)) else t
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fun compile_eq match_name eq =
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  let
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    val (lhs,rhs) = dest_eqs (Logic.strip_imp_concl (strip_alls eq))
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  in
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    compile_lhs match_name lhs (mk_succeed rhs) [] (taken_names eq)
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  end
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(* this is the pattern-matching compiler function *)
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fun compile_eqs match_name eqs =
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  let
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    val (consts, matchers) =
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      ListPair.unzip (map (compile_eq match_name) eqs)
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    val const =
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        case distinct (op =) consts of
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          [n] => n
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        | [] => fixrec_err "no defining equations for function"
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        | _ => fixrec_err "all equations in block must define the same function"
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    val vars =
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        case distinct (op = o pairself length) (map fst matchers) of
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          [vars] => vars
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        | _ => fixrec_err "all equations in block must have the same arity"
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    (* rename so all matchers use same free variables *)
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    fun rename (vs, t) = Term.subst_free (filter_out (op =) (vs ~~ vars)) t
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    val rhs = big_lambdas vars (mk_run (foldr1 mk_mplus (map rename matchers)))
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  in
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    mk_trp (const === rhs)
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  end
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(*************************************************************************)
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(********************** Proving associated theorems **********************)
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(*************************************************************************)
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fun eta_tac i = CONVERSION Thm.eta_conversion i
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fun fixrec_simp_tac ctxt =
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  let
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    val tab = FixrecUnfoldData.get (Context.Proof ctxt)
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    val ss = Simplifier.simpset_of ctxt
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    fun concl t =
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      if Logic.is_all t then concl (snd (Logic.dest_all t))
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      else HOLogic.dest_Trueprop (Logic.strip_imp_concl t)
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    fun tac (t, i) =
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      let
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        val (c, _) =
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            (dest_Const o head_of o chead_of o fst o HOLogic.dest_eq o concl) t
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        val unfold_thm = the (Symtab.lookup tab c)
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        val rule = unfold_thm RS @{thm ssubst_lhs}
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      in
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        CHANGED (rtac rule i THEN eta_tac i THEN asm_simp_tac ss i)
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      end
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  in
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    SUBGOAL (fn ti => the_default no_tac (try tac ti))
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  end
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(* proves a block of pattern matching equations as theorems, using unfold *)
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fun make_simps ctxt (unfold_thm, eqns : (Attrib.binding * term) list) =
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  let
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    val ss = Simplifier.simpset_of ctxt
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    val rule = unfold_thm RS @{thm ssubst_lhs}
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    val tac = rtac rule 1 THEN eta_tac 1 THEN asm_simp_tac ss 1
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    fun prove_term t = Goal.prove ctxt [] [] t (K tac)
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    fun prove_eqn (bind, eqn_t) = (bind, prove_term eqn_t)
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  in
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    map prove_eqn eqns
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  end
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(*************************************************************************)
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(************************* Main fixrec function **************************)
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(*************************************************************************)
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local
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(* code adapted from HOL/Tools/primrec.ML *)
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fun gen_fixrec
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  prep_spec
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  (raw_fixes : (binding * 'a option * mixfix) list)
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  (raw_spec' : (bool * (Attrib.binding * 'b)) list)
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  (lthy : local_theory) =
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  let
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    val (skips, raw_spec) = ListPair.unzip raw_spec'
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    val (fixes : ((binding * typ) * mixfix) list,
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         spec : (Attrib.binding * term) list) =
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          fst (prep_spec raw_fixes raw_spec lthy)
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    val names = map (Binding.name_of o fst o fst) fixes
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    fun check_head name =
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        member (op =) names name orelse
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        fixrec_err ("Illegal equation head. Expected " ^ commas_quote names)
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    val chead_of_spec =
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      chead_of o fst o dest_eqs o Logic.strip_imp_concl o strip_alls o snd
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    fun name_of (Free (n, _)) = tap check_head n
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      | name_of _ = fixrec_err ("unknown term")
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    val all_names = map (name_of o chead_of_spec) spec
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    fun block_of_name n =
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      map_filter
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        (fn (m,eq) => if m = n then SOME eq else NONE)
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        (all_names ~~ (spec ~~ skips))
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    val blocks = map block_of_name names
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    val matcher_tab = FixrecMatchData.get (Proof_Context.theory_of lthy)
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    fun match_name c =
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      case Symtab.lookup matcher_tab c of SOME m => m
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        | NONE => fixrec_err ("unknown pattern constructor: " ^ c)
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    val matches = map (compile_eqs match_name) (map (map (snd o fst)) blocks)
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    val spec' = map (pair Attrib.empty_binding) matches
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    val (lthy, _, _, unfold_thms) =
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      add_fixdefs fixes spec' lthy
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    val blocks' = map (map fst o filter_out snd) blocks
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    val simps : (Attrib.binding * thm) list list =
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      map (make_simps lthy) (unfold_thms ~~ blocks')
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    fun mk_bind n : Attrib.binding =
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     (Binding.qualify true n (Binding.name "simps"), @{attributes [simp]})
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    val simps1 : (Attrib.binding * thm list) list =
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      map (fn (n,xs) => (mk_bind n, map snd xs)) (names ~~ simps)
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    val simps2 : (Attrib.binding * thm list) list =
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      map (apsnd (fn thm => [thm])) (flat simps)
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    val (_, lthy) = lthy
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      |> fold_map Local_Theory.note (simps1 @ simps2)
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  in
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    lthy
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  end
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in
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val add_fixrec = gen_fixrec Specification.check_spec
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val add_fixrec_cmd = gen_fixrec Specification.read_spec
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end (* local *)
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(*************************************************************************)
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(******************************** Parsers ********************************)
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(*************************************************************************)
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val opt_thm_name' : (bool * Attrib.binding) parser =
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  Parse.$$$ "(" -- Parse.$$$ "unchecked" -- Parse.$$$ ")" >> K (true, Attrib.empty_binding)
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    || Parse_Spec.opt_thm_name ":" >> pair false
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val spec' : (bool * (Attrib.binding * string)) parser =
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  opt_thm_name' -- Parse.prop >> (fn ((a, b), c) => (a, (b, c)))
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val alt_specs' : (bool * (Attrib.binding * string)) list parser =
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  let val unexpected = Scan.ahead (Parse.name || Parse.$$$ "[" || Parse.$$$ "(")
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   402
  in Parse.enum1 "|" (spec' --| Scan.option (unexpected -- Parse.!!! (Parse.$$$ "|"))) end
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   403
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   404
val _ =
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   405
  Outer_Syntax.local_theory "fixrec" "define recursive functions (HOLCF)" Keyword.thy_decl
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    (Parse.fixes -- (Parse.where_ |-- Parse.!!! alt_specs')
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   407
      >> (fn (fixes, specs) => add_fixrec_cmd fixes specs))
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val setup =
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  Method.setup @{binding fixrec_simp}
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    (Scan.succeed (SIMPLE_METHOD' o fixrec_simp_tac))
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    "pattern prover for fixrec constants"
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   414
end