src/HOL/Tools/Function/partial_function.ML

partial_function is a declaration command

(*  Title:      HOL/Tools/Function/partial_function.ML
    Author:     Alexander Krauss, TU Muenchen

Partial function definitions based on least fixed points in ccpos.
*)

signature PARTIAL_FUNCTION =
sig
  val setup: theory -> theory
  val init: term -> term -> thm -> declaration

  val add_partial_function: string -> (binding * typ option * mixfix) list ->
    Attrib.binding * term -> local_theory -> local_theory

  val add_partial_function_cmd: string -> (binding * string option * mixfix) list ->
    Attrib.binding * string -> local_theory -> local_theory
end;


structure Partial_Function: PARTIAL_FUNCTION =
struct

(*** Context Data ***)

structure Modes = Generic_Data
(
  type T = ((term * term) * thm) Symtab.table;
  val empty = Symtab.empty;
  val extend = I;
  fun merge (a, b) = Symtab.merge (K true) (a, b);
)

fun init fixp mono fixp_eq phi =
  let
    val term = Morphism.term phi;
    val data' = ((term fixp, term mono), Morphism.thm phi fixp_eq);
    val mode = (* extract mode identifier from morphism prefix! *)
      Binding.prefix_of (Morphism.binding phi (Binding.empty))
      |> map fst |> space_implode ".";
  in
    if mode = "" then I
    else Modes.map (Symtab.update (mode, data'))
  end

val known_modes = Symtab.keys o Modes.get o Context.Proof;
val lookup_mode = Symtab.lookup o Modes.get o Context.Proof;


structure Mono_Rules = Named_Thms
(
  val name = "partial_function_mono";
  val description = "monotonicity rules for partial function definitions";
);


(*** Automated monotonicity proofs ***)

fun strip_cases ctac = ctac #> Seq.map snd;

(*rewrite conclusion with k-th assumtion*)
fun rewrite_with_asm_tac ctxt k =
  Subgoal.FOCUS (fn {context=ctxt', prems, ...} =>
    Local_Defs.unfold_tac ctxt' [nth prems k]) ctxt;

fun dest_case thy t =
  case strip_comb t of
    (Const (case_comb, _), args) =>
      (case Datatype.info_of_case thy case_comb of
         NONE => NONE
       | SOME {case_rewrites, ...} =>
           let
             val lhs = prop_of (hd case_rewrites)
               |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> fst;
             val arity = length (snd (strip_comb lhs));
             val conv = funpow (length args - arity) Conv.fun_conv
               (Conv.rewrs_conv (map mk_meta_eq case_rewrites));
           in
             SOME (nth args (arity - 1), conv)
           end)
  | _ => NONE;

(*split on case expressions*)
val split_cases_tac = Subgoal.FOCUS_PARAMS (fn {context=ctxt, ...} =>
  SUBGOAL (fn (t, i) => case t of
    _ $ (_ $ Abs (_, _, body)) =>
      (case dest_case (ProofContext.theory_of ctxt) body of
         NONE => no_tac
       | SOME (arg, conv) =>
           let open Conv in
              if not (null (loose_bnos arg)) then no_tac
              else ((DETERM o strip_cases o Induct.cases_tac ctxt false [[SOME arg]] NONE [])
                THEN_ALL_NEW (rewrite_with_asm_tac ctxt 0)
                THEN_ALL_NEW etac @{thm thin_rl}
                THEN_ALL_NEW (CONVERSION
                  (params_conv ~1 (fn ctxt' =>
                    arg_conv (arg_conv (abs_conv (K conv) ctxt'))) ctxt))) i
           end)
  | _ => no_tac) 1);

(*monotonicity proof: apply rules + split case expressions*)
fun mono_tac ctxt =
  K (Local_Defs.unfold_tac ctxt [@{thm curry_def}])
  THEN' (TRY o REPEAT_ALL_NEW
   (resolve_tac (Mono_Rules.get ctxt)
     ORELSE' split_cases_tac ctxt));


(*** Auxiliary functions ***)

(*positional instantiation with computed type substitution.
  internal version of  attribute "[of s t u]".*)
fun cterm_instantiate' cts thm =
  let
    val thy = Thm.theory_of_thm thm;
    val vs = rev (Term.add_vars (prop_of thm) [])
      |> map (Thm.cterm_of thy o Var);
  in
    cterm_instantiate (zip_options vs cts) thm
  end;

(*Returns t $ u, but instantiates the type of t to make the
application type correct*)
fun apply_inst ctxt t u =
  let
    val thy = ProofContext.theory_of ctxt;
    val T = domain_type (fastype_of t);
    val T' = fastype_of u;
    val subst = Type.typ_match (Sign.tsig_of thy) (T, T') Vartab.empty
      handle Type.TYPE_MATCH => raise TYPE ("apply_inst", [T, T'], [t, u])
  in
    map_types (Envir.norm_type subst) t $ u
  end;

fun head_conv cv ct =
  if can Thm.dest_comb ct then Conv.fun_conv (head_conv cv) ct else cv ct;


(*** currying transformation ***)

fun curry_const (A, B, C) =
  Const (@{const_name Product_Type.curry},
    [HOLogic.mk_prodT (A, B) --> C, A, B] ---> C);

fun mk_curry f =
  case fastype_of f of
    Type ("fun", [Type (_, [S, T]), U]) =>
      curry_const (S, T, U) $ f
  | T => raise TYPE ("mk_curry", [T], [f]);

(* iterated versions. Nonstandard left-nested tuples arise naturally
from "split o split o split"*)
fun curry_n arity = funpow (arity - 1) mk_curry;
fun uncurry_n arity = funpow (arity - 1) HOLogic.mk_split;

val curry_uncurry_ss = HOL_basic_ss addsimps
  [@{thm Product_Type.curry_split}, @{thm Product_Type.split_curry}]


(*** partial_function definition ***)

fun gen_add_partial_function prep mode fixes_raw eqn_raw lthy =
  let
    val ((fixp, mono), fixp_eq) = the (lookup_mode lthy mode)
      handle Option.Option => error (cat_lines ["Unknown mode " ^ quote mode ^ ".",
        "Known modes are " ^ commas_quote (known_modes lthy) ^ "."]);

    val ((fixes, [(eq_abinding, eqn)]), _) = prep fixes_raw [eqn_raw] lthy;
    val (_, _, plain_eqn) = Function_Lib.dest_all_all_ctx lthy eqn;

    val ((f_binding, fT), mixfix) = the_single fixes;
    val fname = Binding.name_of f_binding;

    val cert = cterm_of (ProofContext.theory_of lthy);
    val (lhs, rhs) = HOLogic.dest_eq (HOLogic.dest_Trueprop plain_eqn);
    val (head, args) = strip_comb lhs;
    val F = fold_rev lambda (head :: args) rhs;

    val arity = length args;
    val (aTs, bTs) = chop arity (binder_types fT);

    val tupleT = foldl1 HOLogic.mk_prodT aTs;
    val fT_uc = tupleT :: bTs ---> body_type fT;
    val f_uc = Var ((fname, 0), fT_uc);
    val x_uc = Var (("x", 0), tupleT);
    val uncurry = lambda head (uncurry_n arity head);
    val curry = lambda f_uc (curry_n arity f_uc);

    val F_uc =
      lambda f_uc (uncurry_n arity (F $ curry_n arity f_uc));

    val mono_goal = apply_inst lthy mono (lambda f_uc (F_uc $ f_uc $ x_uc))
      |> HOLogic.mk_Trueprop
      |> Logic.all x_uc;

    val mono_thm = Goal.prove_internal [] (cert mono_goal)
        (K (mono_tac lthy 1))
      |> Thm.forall_elim (cert x_uc);

    val f_def_rhs = curry_n arity (apply_inst lthy fixp F_uc);
    val f_def_binding = Binding.conceal (Binding.name (Thm.def_name fname));
    val ((f, (_, f_def)), lthy') = Local_Theory.define
      ((f_binding, mixfix), ((f_def_binding, []), f_def_rhs)) lthy;

    val eqn = HOLogic.mk_eq (list_comb (f, args),
        Term.betapplys (F, f :: args))
      |> HOLogic.mk_Trueprop;

    val unfold =
      (cterm_instantiate' (map (SOME o cert) [uncurry, F, curry]) fixp_eq
        OF [mono_thm, f_def])
      |> Tactic.rule_by_tactic lthy (Simplifier.simp_tac curry_uncurry_ss 1);

    val rec_rule = let open Conv in
      Goal.prove lthy' (map (fst o dest_Free) args) [] eqn (fn _ =>
        CONVERSION ((arg_conv o arg1_conv o head_conv o rewr_conv) (mk_meta_eq unfold)) 1
        THEN rtac @{thm refl} 1) end;
  in
    lthy'
    |> Local_Theory.note (eq_abinding, [rec_rule])
    |-> (fn (_, rec') =>
      Spec_Rules.add Spec_Rules.Equational ([f], rec')
      #> Local_Theory.note ((Binding.qualify true fname (Binding.name "simps"), []), rec') #> snd)
  end;

val add_partial_function = gen_add_partial_function Specification.check_spec;
val add_partial_function_cmd = gen_add_partial_function Specification.read_spec;

val mode = Parse.$$$ "(" |-- Parse.xname --| Parse.$$$ ")";

val _ = Outer_Syntax.local_theory
  "partial_function" "define partial function" Keyword.thy_decl
  ((mode -- (Parse.fixes -- (Parse.where_ |-- Parse_Spec.spec)))
     >> (fn (mode, (fixes, spec)) => add_partial_function_cmd mode fixes spec));


val setup = Mono_Rules.setup;

end