src/HOL/Tools/function_package/mutual.ML

Induction rules have schematic variables again.

(*  Title:      HOL/Tools/function_package/mutual.ML
    ID:         $Id$
    Author:     Alexander Krauss, TU Muenchen

A package for general recursive function definitions.
Tools for mutual recursive definitions.
*)

signature FUNDEF_MUTUAL =
sig

  val prepare_fundef_mutual : ((string * typ) * mixfix) list
                              -> term list
                              -> string (* default, unparsed term *)
                              -> local_theory
                              -> ((FundefCommon.mutual_info * string * FundefCommon.prep_result) * local_theory)


  val mk_partial_rules_mutual : Proof.context -> FundefCommon.mutual_info -> FundefCommon.prep_result -> thm ->
                                FundefCommon.fundef_mresult

  val sort_by_function : FundefCommon.mutual_info -> string list -> 'a list -> 'a list list

end


structure FundefMutual: FUNDEF_MUTUAL =
struct

open FundefCommon

(* Theory dependencies *)
val sum_case_rules = thms "Datatype.sum.cases"
val split_apply = thm "Product_Type.split"



fun mutual_induct_Pnames n =
    if n < 5 then fst (chop n ["P","Q","R","S"])
    else map (fn i => "P" ^ string_of_int i) (1 upto n)


fun open_all_all (Const ("all", _) $ Abs (n, T, b)) = apfst (cons (n, T)) (open_all_all b)
  | open_all_all t = ([], t)



(* Builds a curried clause description in abstracted form *)
fun split_def ctxt fnames geq arities =
    let
      fun input_error msg = error (cat_lines [msg, ProofContext.string_of_term ctxt geq])

      val (qs, imp) = open_all_all geq

      val gs = Logic.strip_imp_prems imp
      val eq = Logic.strip_imp_concl imp

      val (f_args, rhs) = HOLogic.dest_eq (HOLogic.dest_Trueprop eq)
      val (head, args) = strip_comb f_args

      val invalid_head_msg = "Head symbol of left hand side must be " ^ plural "" "one out of " fnames ^ commas_quote fnames
      val fname = fst (dest_Free head)
          handle TERM _ => input_error invalid_head_msg

      val _ = if fname mem fnames then ()
              else input_error invalid_head_msg

      fun add_bvs t is = add_loose_bnos (t, 0, is)
      val rvs = (add_bvs rhs [] \\ fold add_bvs args [])
                  |> map (fst o nth (rev qs))

      val _ = if null rvs then ()
              else input_error ("Variable" ^ plural " " "s " rvs ^ commas_quote rvs
                                ^ " occur" ^ plural "s" "" rvs ^ " on right hand side only:")

      val _ = (fold o fold_aterms)
                 (fn Free (n, _) => if n mem fnames
                                    then input_error "Recursive Calls not allowed in premises:"
                                    else I
                   | _ => I) gs ()

      val k = length args

      val arities' = case Symtab.lookup arities fname of
                   NONE => Symtab.update (fname, k) arities
                 | SOME i => if (i <> k)
                             then input_error ("Function " ^ quote fname ^ " has different numbers of arguments in different equations")
                             else arities
    in
        ((fname, qs, gs, args, rhs), arities')
    end

fun get_part fname =
    the o find_first (fn (MutualPart {fvar=(n,_), ...}) => n = fname)

(* FIXME *)
fun mk_prod_abs e (t1, t2) =
    let
      val bTs = rev (map snd e)
      val T1 = fastype_of1 (bTs, t1)
      val T2 = fastype_of1 (bTs, t2)
    in
      HOLogic.pair_const T1 T2 $ t1 $ t2
    end;


fun analyze_eqs ctxt fs eqs =
    let
        val fnames = map fst fs
        val (fqgars, arities) = fold_map (split_def ctxt fnames) eqs Symtab.empty

        fun curried_types (fname, fT) =
            let
              val k = the_default 1 (Symtab.lookup arities fname)
              val (caTs, uaTs) = chop k (binder_types fT)
            in
                (caTs, uaTs ---> body_type fT)
            end

        val (caTss, resultTs) = split_list (map curried_types fs)
        val argTs = map (foldr1 HOLogic.mk_prodT) caTss

        val (RST,streeR, pthsR) = SumTools.mk_tree_distinct resultTs
        val (ST, streeA, pthsA) = SumTools.mk_tree argTs

        val def_name = foldr1 (fn (a,b) => a ^ "_" ^ b) (map Sign.base_name fnames)
        val fsum_type = ST --> RST

        val ([fsum_var_name], _) = Variable.add_fixes [ def_name ^ "_sum" ] ctxt
        val fsum_var = (fsum_var_name, fsum_type)

        fun define (fvar as (n, T)) caTs pthA pthR =
            let
                val vars = map_index (fn (i,T) => Free ("x" ^ string_of_int i, T)) caTs (* FIXME: Bind xs properly *)

                val f_exp = SumTools.mk_proj streeR pthR (Free fsum_var $ SumTools.mk_inj streeA pthA (foldr1 HOLogic.mk_prod vars))
                val def = Term.abstract_over (Free fsum_var, fold_rev lambda vars f_exp)

                val rew = (n, fold_rev lambda vars f_exp)
            in
                (MutualPart {fvar=fvar,cargTs=caTs,pthA=pthA,pthR=pthR,f_def=def,f=NONE,f_defthm=NONE}, rew)
            end

        val (parts, rews) = split_list (map4 define fs caTss pthsA pthsR)

        fun convert_eqs (f, qs, gs, args, rhs) =
            let
              val MutualPart {pthA, pthR, ...} = get_part f parts
            in
              (qs, gs, SumTools.mk_inj streeA pthA (foldr1 (mk_prod_abs qs) args),
               SumTools.mk_inj streeR pthR (replace_frees rews rhs)
                               |> Envir.norm_term (Envir.empty 0))
            end

        val qglrs = map convert_eqs fqgars
    in
        Mutual {defname=def_name,fsum_var=fsum_var, ST=ST, RST=RST, streeA=streeA, streeR=streeR,
                parts=parts, fqgars=fqgars, qglrs=qglrs, fsum=NONE}
    end




fun define_projections fixes mutual fsum lthy =
    let
      fun def ((MutualPart {fvar=(fname, fT), cargTs, pthA, pthR, f_def, ...}), (_, mixfix)) lthy =
          let
            val ((f, (_, f_defthm)), lthy') = LocalTheory.def ((fname, mixfix),
                                                               ((fname ^ "_def", []), Term.subst_bound (fsum, f_def)))
                                                              lthy
          in
            (MutualPart {fvar=(fname, fT), cargTs=cargTs, pthA=pthA, pthR=pthR, f_def=f_def,
                        f=SOME f, f_defthm=SOME f_defthm },
             lthy')
          end

      val Mutual { defname, fsum_var, ST, RST, streeA, streeR, parts, fqgars, qglrs, ... } = mutual
      val (parts', lthy') = fold_map def (parts ~~ fixes) lthy
    in
      (Mutual { defname=defname, fsum_var=fsum_var, ST=ST, RST=RST, streeA=streeA, streeR=streeR, parts=parts',
                fqgars=fqgars, qglrs=qglrs, fsum=SOME fsum },
       lthy')
    end


fun prepare_fundef_mutual fixes eqss default lthy =
    let
        val mutual = analyze_eqs lthy (map fst fixes) eqss
        val Mutual {defname, fsum_var=(n, T), qglrs, ...} = mutual

        val (prep_result, fsum, lthy') =
            FundefPrep.prepare_fundef defname (n, T, NoSyn) qglrs default lthy

        val (mutual', lthy'') = define_projections fixes mutual fsum lthy'
    in
      ((mutual', defname, prep_result), lthy'')
    end


(* Beta-reduce both sides of a meta-equality *)
fun beta_norm_eq thm =
    let
        val (lhs, rhs) = dest_equals (cprop_of thm)
        val lhs_conv = beta_conversion false lhs
        val rhs_conv = beta_conversion false rhs
    in
        transitive (symmetric lhs_conv) (transitive thm rhs_conv)
    end

fun beta_reduce thm = Thm.equal_elim (Thm.beta_conversion true (cprop_of thm)) thm


fun in_context ctxt (f, pre_qs, pre_gs, pre_args, pre_rhs) F =
    let
      val thy = ProofContext.theory_of ctxt

      val oqnames = map fst pre_qs
      val (qs, ctxt') = Variable.variant_fixes oqnames ctxt
                                           |>> map2 (fn (_, T) => fn n => Free (n, T)) pre_qs

      fun inst t = subst_bounds (rev qs, t)
      val gs = map inst pre_gs
      val args = map inst pre_args
      val rhs = inst pre_rhs

      val cqs = map (cterm_of thy) qs
      val ags = map (assume o cterm_of thy) gs

      val import = fold forall_elim cqs
                   #> fold implies_elim_swp ags

      val export = fold_rev (implies_intr o cprop_of) ags
                   #> fold_rev forall_intr_rename (oqnames ~~ cqs)
    in
      F (f, qs, gs, args, rhs) import export
    end


fun recover_mutual_psimp thy RST streeR all_f_defs parts (f, _, _, args, _) import (export : thm -> thm) sum_psimp_eq =
    let
      val (MutualPart {f_defthm=SOME f_def, pthR, ...}) = get_part f parts

      val psimp = import sum_psimp_eq
      val (simp, restore_cond) = case cprems_of psimp of
                                   [] => (psimp, I)
                                 | [cond] => (implies_elim psimp (assume cond), implies_intr cond)
                                 | _ => sys_error "Too many conditions"

      val x = Free ("x", RST)

      val f_def_inst = fold (fn arg => fn thm => combination thm (reflexive (cterm_of thy arg))) args (Thm.freezeT f_def) (* FIXME *)
                            |> beta_reduce
    in
      reflexive (cterm_of thy (lambda x (SumTools.mk_proj streeR pthR x)))  (*  PR(x) == PR(x) *)
                |> (fn it => combination it (simp RS eq_reflection))
                |> beta_norm_eq (*  PR(S(I(as))) == PR(IR(...)) *)
                |> transitive f_def_inst (*  f ... == PR(IR(...)) *)
                |> simplify (HOL_basic_ss addsimps [SumTools.projl_inl, SumTools.projr_inr]) (*  f ... == ... *)
                |> simplify (HOL_basic_ss addsimps all_f_defs) (*  f ... == ... *)
                |> (fn it => it RS meta_eq_to_obj_eq)
                |> restore_cond
                |> export
    end


(* FIXME HACK *)
fun mk_applied_form ctxt caTs thm =
    let
      val thy = ProofContext.theory_of ctxt
      val xs = map_index (fn (i,T) => cterm_of thy (Free ("x" ^ string_of_int i, T))) caTs (* FIXME: Bind xs properly *)
    in
      fold (fn x => fn thm => combination thm (reflexive x)) xs thm
           |> beta_reduce
           |> fold_rev forall_intr xs
           |> forall_elim_vars 0
    end


fun mutual_induct_rules thy induct all_f_defs (Mutual {RST, parts, streeA, ...}) =
    let
      val newPs = map2 (fn Pname => fn MutualPart {cargTs, ...} => 
                                   Free (Pname, cargTs ---> HOLogic.boolT))
                       (mutual_induct_Pnames (length parts))
                       parts

        fun mk_P (MutualPart {cargTs, ...}) P =
            let
                val avars = map_index (fn (i,T) => Var (("a", i), T)) cargTs
                val atup = foldr1 HOLogic.mk_prod avars
            in
                tupled_lambda atup (list_comb (P, avars))
            end

        val Ps = map2 mk_P parts newPs
        val case_exp = SumTools.mk_sumcases streeA HOLogic.boolT Ps

        val induct_inst =
            forall_elim (cterm_of thy case_exp) induct
                        |> full_simplify (HOL_basic_ss addsimps (split_apply :: sum_case_rules))
                        |> full_simplify (HOL_basic_ss addsimps all_f_defs)

        fun mk_proj rule (MutualPart {cargTs, pthA, ...}) =
            let
                val afs = map_index (fn (i,T) => Free ("a" ^ string_of_int i, T)) cargTs
                val inj = SumTools.mk_inj streeA pthA (foldr1 HOLogic.mk_prod afs)
            in
                rule
                    |> forall_elim (cterm_of thy inj)
                    |> full_simplify (HOL_basic_ss addsimps (split_apply :: sum_case_rules))
                    |> fold_rev (forall_intr o cterm_of thy) (afs @ newPs)
            end

    in
      map (mk_proj induct_inst) parts
    end





fun mk_partial_rules_mutual lthy (m as Mutual {RST, parts, streeR, fqgars, ...}) data prep_result =
    let
      val thy = ProofContext.theory_of lthy

      (* FIXME !? *)
      val expand = Assumption.export false lthy (LocalTheory.target_of lthy);
      val expand_term = Drule.term_rule thy expand;

      val result = FundefProof.mk_partial_rules thy data prep_result
      val FundefResult {f, G, R, completeness, psimps, subset_pinduct,simple_pinduct,total_intro,dom_intros} = result

      val all_f_defs = map (fn MutualPart {f_defthm = SOME f_def, cargTs, ...} =>
                               mk_applied_form lthy cargTs (symmetric (Thm.freezeT f_def)))
                           parts

      fun mk_mpsimp fqgar sum_psimp =
          in_context lthy fqgar (recover_mutual_psimp thy RST streeR all_f_defs parts) sum_psimp

      val mpsimps = map2 mk_mpsimp fqgars psimps

      val minducts = mutual_induct_rules thy simple_pinduct all_f_defs m
      val termination = full_simplify (HOL_basic_ss addsimps all_f_defs) total_intro
    in
      FundefMResult { f=expand_term f, G=expand_term G, R=expand_term R,
                                  psimps=map expand mpsimps, subset_pinducts=[expand subset_pinduct], simple_pinducts=map expand minducts,
                                  cases=expand completeness, termination=expand termination,
                      domintros=map expand dom_intros }
    end



(* puts an object in the "right bucket" *)
fun store_grouped P x [] = []
  | store_grouped P x ((l, xs)::bs) =
    if P (x, l) then ((l, x::xs)::bs) else ((l, xs)::store_grouped P x bs)

fun sort_by_function (Mutual {fqgars, ...}) names xs =
      fold_rev (store_grouped (eq_str o apfst fst))  (* fill *)
               (map name_of_fqgar fqgars ~~ xs)      (* the name-thm pairs *)
               (map (rpair []) names)                (* in the empty buckets labeled with names *)

         |> map (snd #> map snd)                     (* and remove the labels afterwards *)

end