(*  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: string -> term -> term -> thm -> thm option -> 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 ***)

 datatype setup_data = Setup_Data of 

  {fixp: term,

   mono: term,

   fixp_eq: thm,

   fixp_induct: thm option};

 structure Modes = Generic_Data

 (

   type T = setup_data Symtab.table;

   val empty = Symtab.empty;

   val extend = I;

   fun merge data = Symtab.merge (K true) data;

 )

 fun init mode fixp mono fixp_eq fixp_induct phi =

   let

     val term = Morphism.term phi;

     val thm = Morphism.thm phi;

     val data' = Setup_Data 

       {fixp=term fixp, mono=term mono, fixp_eq=thm fixp_eq,

        fixp_induct=Option.map thm fixp_induct};

   in

     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 = @{binding 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 (Proof_Context.theory_of ctxt) body of

          NONE => no_tac

        | SOME (arg, conv) =>

            let open Conv in

               if Term.is_open 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 = Proof_Context.theory_of ctxt;

     val T = domain_type (fastype_of t);

     val T' = fastype_of u;

     val subst = Sign.typ_match 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}]

 val split_conv_ss = HOL_basic_ss addsimps

   [@{thm Product_Type.split_conv}];

 fun mk_curried_induct args ctxt ccurry cuncurry rule =

   let

     val cert = Thm.cterm_of (Proof_Context.theory_of ctxt)

     val ([P], ctxt') = Variable.variant_fixes ["P"] ctxt

     val split_paired_all_conv =

       Conv.every_conv (replicate (length args - 1) (Conv.rewr_conv @{thm split_paired_all}))

     val split_params_conv = 

       Conv.params_conv ~1 (fn ctxt' =>

         Conv.implies_conv split_paired_all_conv Conv.all_conv)

     val inst_rule =

       cterm_instantiate' [SOME cuncurry, NONE, SOME ccurry] rule

     val plain_resultT = 

       Thm.prop_of inst_rule |> Logic.strip_imp_concl |> HOLogic.dest_Trueprop

       |> Term.head_of |> Term.dest_Var |> snd |> range_type |> domain_type

     val PT = map (snd o dest_Free) args ---> plain_resultT --> HOLogic.boolT

     val x_inst = cert (foldl1 HOLogic.mk_prod args)

     val P_inst = cert (uncurry_n (length args) (Free (P, PT)))

     val inst_rule' = inst_rule

       |> Tactic.rule_by_tactic ctxt

         (Simplifier.simp_tac curry_uncurry_ss 4

          THEN Simplifier.simp_tac curry_uncurry_ss 3

          THEN CONVERSION (split_params_conv ctxt

            then_conv (Conv.forall_conv (K split_paired_all_conv) ctxt)) 3)

       |> Drule.instantiate' [] [NONE, NONE, SOME P_inst, SOME x_inst]

       |> Simplifier.full_simplify split_conv_ss

       |> singleton (Variable.export ctxt' ctxt)

   in

     inst_rule'

   end;

 (*** partial_function definition ***)

 fun gen_add_partial_function prep mode fixes_raw eqn_raw lthy =

   let

     val setup_data = the (lookup_mode lthy mode)

       handle Option.Option => error (cat_lines ["Unknown mode " ^ quote mode ^ ".",

         "Known modes are " ^ commas_quote (known_modes lthy) ^ "."]);

     val Setup_Data {fixp, mono, fixp_eq, fixp_induct} = setup_data;

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

     val ((_, plain_eqn), args_ctxt) = Variable.focus eqn lthy;

     val ((f_binding, fT), mixfix) = the_single fixes;

     val fname = Binding.name_of f_binding;

     val cert = cterm_of (Proof_Context.theory_of lthy);

     val (lhs, rhs) = HOLogic.dest_eq (HOLogic.dest_Trueprop plain_eqn);

     val (head, args) = strip_comb lhs;

     val argnames = map (fst o dest_Free) args;

     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 mk_raw_induct =

       mk_curried_induct args args_ctxt (cert curry) (cert uncurry)

       #> singleton (Variable.export args_ctxt lthy)

       #> (fn thm => cterm_instantiate' [SOME (cert F)] thm OF [mono_thm, f_def])

       #> Drule.rename_bvars' (map SOME (fname :: argnames @ argnames))

     val raw_induct = Option.map mk_raw_induct fixp_induct

     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)

     |> (case raw_induct of NONE => I | SOME thm =>

          Local_Theory.note ((Binding.qualify true fname (Binding.name "raw_induct"), []), [thm]) #> 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 = @{keyword "("} |-- Parse.xname --| @{keyword ")"};

 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