File tfl.ML (Isabelle2005: October 2005)

# File tfl.ML

    (*  Title:      TFL/tfl.ML
        ID:         $Id: tfl.ML,v 1.15 2005/09/17 16:11:25 wenzelm Exp $
        Author:     Konrad Slind, Cambridge University Computer Laboratory
        Copyright   1997  University of Cambridge

    First part of main module.
    *)

    signature PRIM =
    sig
      val trace: bool ref
      val trace_thms: string -> thm list -> unit
      val trace_cterms: string -> cterm list -> unit
      type pattern
      val mk_functional: theory -> term list -> {functional: term, pats: pattern list}
      val wfrec_definition0: theory -> string -> term -> term -> theory * thm
      val post_definition: thm list -> theory * (thm * pattern list) ->
       {theory: theory,
        rules: thm,
        rows: int list,
        TCs: term list list,
        full_pats_TCs: (term * term list) list}
      val wfrec_eqns: theory -> xstring -> thm list -> term list ->
       {WFR: term,
        SV: term list,
        proto_def: term,
        extracta: (thm * term list) list,
        pats: pattern list}
      val lazyR_def: theory -> xstring -> thm list -> term list ->
       {theory: theory,
        rules: thm,
        R: term,
        SV: term list,
        full_pats_TCs: (term * term list) list,
        patterns : pattern list}
      val mk_induction: theory ->
        {fconst: term, R: term, SV: term list, pat_TCs_list: (term * term list) list} -> thm
      val postprocess: bool -> {wf_tac: tactic, terminator: tactic, simplifier: cterm -> thm}
        -> theory -> {rules: thm, induction: thm, TCs: term list list}
        -> {rules: thm, induction: thm, nested_tcs: thm list}
    end;

    structure Prim: PRIM =
    struct

    val trace = ref false;

    structure R = Rules;
    structure S = USyntax;
    structure U = Utils;

    fun TFL_ERR func mesg = U.ERR {module = "Tfl", func = func, mesg = mesg};

    val concl = #2 o R.dest_thm;
    val hyp = #1 o R.dest_thm;

    val list_mk_type = U.end_itlist (curry (op -->));

    fun enumerate xs = ListPair.zip(xs, 0 upto (length xs - 1));

    fun front_last [] = raise TFL_ERR "front_last" "empty list"
      | front_last [x] = ([],x)
      | front_last (h::t) =
         let val (pref,x) = front_last t
         in
            (h::pref,x)
         end;

    (*---------------------------------------------------------------------------
     * The next function is common to pattern-match translation and
     * proof of completeness of cases for the induction theorem.
     *
     * The curried function "gvvariant" returns a function to generate distinct
     * variables that are guaranteed not to be in names.  The names of
     * the variables go u, v, ..., z, aa, ..., az, ...  The returned
     * function contains embedded refs!
     *---------------------------------------------------------------------------*)
    fun gvvariant names =
      let val slist = ref names
          val vname = ref "u"
          fun new() =
             if !vname mem_string (!slist)
             then (vname := Symbol.bump_string (!vname);  new())
             else (slist := !vname :: !slist;  !vname)
      in
      fn ty => Free(new(), ty)
      end;

    (*---------------------------------------------------------------------------
     * Used in induction theorem production. This is the simple case of
     * partitioning up pattern rows by the leading constructor.
     *---------------------------------------------------------------------------*)
    fun ipartition gv (constructors,rows) =
      let fun pfail s = raise TFL_ERR "partition.part" s
          fun part {constrs = [],   rows = [],   A} = rev A
            | part {constrs = [],   rows = _::_, A} = pfail"extra cases in defn"
            | part {constrs = _::_, rows = [],   A} = pfail"cases missing in defn"
            | part {constrs = c::crst, rows,     A} =
              let val (c, T) = dest_Const c
                  val L = binder_types T
                  val (in_group, not_in_group) =
                   fold_rev (fn (row as (p::rst, rhs)) =>
                             fn (in_group,not_in_group) =>
                      let val (pc,args) = S.strip_comb p
                      in if (#1(dest_Const pc) = c)
                         then ((args@rst, rhs)::in_group, not_in_group)
                         else (in_group, row::not_in_group)
                      end)      rows ([],[])
                  val col_types = U.take type_of (length L, #1(hd in_group))
              in
              part{constrs = crst, rows = not_in_group,
                   A = {constructor = c,
                        new_formals = map gv col_types,
                        group = in_group}::A}
              end
      in part{constrs = constructors, rows = rows, A = []}
      end;

    (*---------------------------------------------------------------------------
     * Each pattern carries with it a tag (i,b) where
     * i is the clause it came from and
     * b=true indicates that clause was given by the user
     * (or is an instantiation of a user supplied pattern)
     * b=false --> i = ~1
     *---------------------------------------------------------------------------*)

    type pattern = term * (int * bool)

    fun pattern_map f (tm,x) = (f tm, x);

    fun pattern_subst theta = pattern_map (subst_free theta);

    val pat_of = fst;
    fun row_of_pat x = fst (snd x);
    fun given x = snd (snd x);

    (*---------------------------------------------------------------------------
     * Produce an instance of a constructor, plus genvars for its arguments.
     *---------------------------------------------------------------------------*)
    fun fresh_constr ty_match colty gv c =
      let val (_,Ty) = dest_Const c
          val L = binder_types Ty
          and ty = body_type Ty
          val ty_theta = ty_match ty colty
          val c' = S.inst ty_theta c
          val gvars = map (S.inst ty_theta o gv) L
      in (c', gvars)
      end;

    (*---------------------------------------------------------------------------
     * Goes through a list of rows and picks out the ones beginning with a
     * pattern with constructor = name.
     *---------------------------------------------------------------------------*)
    fun mk_group name rows =
      fold_rev (fn (row as ((prfx, p::rst), rhs)) =>
                fn (in_group,not_in_group) =>
                   let val (pc,args) = S.strip_comb p
                   in if ((#1 (Term.dest_Const pc) = name) handle TERM _ => false)
                      then (((prfx,args@rst), rhs)::in_group, not_in_group)
                      else (in_group, row::not_in_group) end)
          rows ([],[]);

    (*---------------------------------------------------------------------------
     * Partition the rows. Not efficient: we should use hashing.
     *---------------------------------------------------------------------------*)
    fun partition _ _ (_,_,_,[]) = raise TFL_ERR "partition" "no rows"
      | partition gv ty_match
                  (constructors, colty, res_ty, rows as (((prfx,_),_)::_)) =
    let val fresh = fresh_constr ty_match colty gv
         fun part {constrs = [],      rows, A} = rev A
           | part {constrs = c::crst, rows, A} =
             let val (c',gvars) = fresh c
                 val (in_group, not_in_group) = mk_group (#1 (dest_Const c')) rows
                 val in_group' =
                     if (null in_group)  (* Constructor not given *)
                     then [((prfx, #2(fresh c)), (S.ARB res_ty, (~1,false)))]
                     else in_group
             in
             part{constrs = crst,
                  rows = not_in_group,
                  A = {constructor = c',
                       new_formals = gvars,
                       group = in_group'}::A}
             end
    in part{constrs=constructors, rows=rows, A=[]}
    end;

    (*---------------------------------------------------------------------------
     * Misc. routines used in mk_case
     *---------------------------------------------------------------------------*)

    fun mk_pat (c,l) =
      let val L = length (binder_types (type_of c))
          fun build (prfx,tag,plist) =
              let val args   = Library.take (L,plist)
                  and plist' = Library.drop(L,plist)
              in (prfx,tag,list_comb(c,args)::plist') end
      in map build l end;

    fun v_to_prfx (prfx, v::pats) = (v::prfx,pats)
      | v_to_prfx _ = raise TFL_ERR "mk_case" "v_to_prfx";

    fun v_to_pats (v::prfx,tag, pats) = (prfx, tag, v::pats)
      | v_to_pats _ = raise TFL_ERR "mk_case" "v_to_pats";

    (*----------------------------------------------------------------------------
     * Translation of pattern terms into nested case expressions.
     *
     * This performs the translation and also builds the full set of patterns.
     * Thus it supports the construction of induction theorems even when an
     * incomplete set of patterns is given.
     *---------------------------------------------------------------------------*)

    fun mk_case ty_info ty_match usednames range_ty =
     let
     fun mk_case_fail s = raise TFL_ERR "mk_case" s
     val fresh_var = gvvariant usednames
     val divide = partition fresh_var ty_match
     fun expand constructors ty ((_,[]), _) = mk_case_fail"expand_var_row"
       | expand constructors ty (row as ((prfx, p::rst), rhs)) =
           if (is_Free p)
           then let val fresh = fresh_constr ty_match ty fresh_var
                    fun expnd (c,gvs) =
                      let val capp = list_comb(c,gvs)
                      in ((prfx, capp::rst), pattern_subst[(p,capp)] rhs)
                      end
                in map expnd (map fresh constructors)  end
           else [row]
     fun mk{rows=[],...} = mk_case_fail"no rows"
       | mk{path=[], rows = ((prfx, []), (tm,tag))::_} =  (* Done *)
            ([(prfx,tag,[])], tm)
       | mk{path=[], rows = _::_} = mk_case_fail"blunder"
       | mk{path as u::rstp, rows as ((prfx, []), rhs)::rst} =
            mk{path = path,
               rows = ((prfx, [fresh_var(type_of u)]), rhs)::rst}
       | mk{path = u::rstp, rows as ((_, p::_), _)::_} =
         let val (pat_rectangle,rights) = ListPair.unzip rows
             val col0 = map(hd o #2) pat_rectangle
         in
         if (forall is_Free col0)
         then let val rights' = map (fn(v,e) => pattern_subst[(v,u)] e)
                                    (ListPair.zip (col0, rights))
                  val pat_rectangle' = map v_to_prfx pat_rectangle
                  val (pref_patl,tm) = mk{path = rstp,
                                          rows = ListPair.zip (pat_rectangle',
                                                               rights')}
              in (map v_to_pats pref_patl, tm)
              end
         else
         let val pty as Type (ty_name,_) = type_of p
         in
         case (ty_info ty_name)
         of NONE => mk_case_fail("Not a known datatype: "^ty_name)
          | SOME{case_const,constructors} =>
            let
                val case_const_name = #1(dest_Const case_const)
                val nrows = List.concat (map (expand constructors pty) rows)
                val subproblems = divide(constructors, pty, range_ty, nrows)
                val groups      = map #group subproblems
                and new_formals = map #new_formals subproblems
                and constructors' = map #constructor subproblems
                val news = map (fn (nf,rows) => {path = nf@rstp, rows=rows})
                               (ListPair.zip (new_formals, groups))
                val rec_calls = map mk news
                val (pat_rect,dtrees) = ListPair.unzip rec_calls
                val case_functions = map S.list_mk_abs
                                      (ListPair.zip (new_formals, dtrees))
                val types = map type_of (case_functions@[u]) @ [range_ty]
                val case_const' = Const(case_const_name, list_mk_type types)
                val tree = list_comb(case_const', case_functions@[u])
                val pat_rect1 = List.concat
                                  (ListPair.map mk_pat (constructors', pat_rect))
            in (pat_rect1,tree)
            end
         end end
     in mk
     end;

    (* Repeated variable occurrences in a pattern are not allowed. *)
    fun FV_multiset tm =
       case (S.dest_term tm)
         of S.VAR{Name = c, Ty = T} => [Free(c, T)]
          | S.CONST _ => []
          | S.COMB{Rator, Rand} => FV_multiset Rator @ FV_multiset Rand
          | S.LAMB _ => raise TFL_ERR "FV_multiset" "lambda";

    fun no_repeat_vars thy pat =
     let fun check [] = true
           | check (v::rst) =
             if mem_term (v,rst) then
                raise TFL_ERR "no_repeat_vars"
                              (quote (#1 (dest_Free v)) ^
                              " occurs repeatedly in the pattern " ^
                              quote (string_of_cterm (Thry.typecheck thy pat)))
             else check rst
     in check (FV_multiset pat)
     end;

    fun dest_atom (Free p) = p
      | dest_atom (Const p) = p
      | dest_atom  _ = raise TFL_ERR "dest_atom" "function name not an identifier";

    fun same_name (p,q) = #1(dest_atom p) = #1(dest_atom q);

    local fun mk_functional_err s = raise TFL_ERR "mk_functional" s
          fun single [_$_] =
                  mk_functional_err "recdef does not allow currying"
            | single [f] = f
            | single fs  =
                  (*multiple function names?*)
                  if length (gen_distinct same_name fs) < length fs
                  then mk_functional_err
                       "The function being declared appears with multiple types"
                  else mk_functional_err
                       (Int.toString (length fs) ^
                        " distinct function names being declared")
    in
    fun mk_functional thy clauses =
     let val (L,R) = ListPair.unzip (map HOLogic.dest_eq clauses
                       handle TERM _ => raise TFL_ERR "mk_functional"
                            "recursion equations must use the = relation")
         val (funcs,pats) = ListPair.unzip (map (fn (t$u) =>(t,u)) L)
         val atom = single (gen_distinct (op aconv) funcs)
         val (fname,ftype) = dest_atom atom
         val dummy = map (no_repeat_vars thy) pats
         val rows = ListPair.zip (map (fn x => ([]:term list,[x])) pats,
                                  map (fn (t,i) => (t,(i,true))) (enumerate R))
         val names = foldr add_term_names [] R
         val atype = type_of(hd pats)
         and aname = variant names "a"
         val a = Free(aname,atype)
         val ty_info = Thry.match_info thy
         val ty_match = Thry.match_type thy
         val range_ty = type_of (hd R)
         val (patts, case_tm) = mk_case ty_info ty_match (aname::names) range_ty
                                        {path=[a], rows=rows}
         val patts1 = map (fn (_,tag,[pat]) => (pat,tag)) patts
              handle Match => mk_functional_err "error in pattern-match translation"
         val patts2 = Library.sort (Library.int_ord o Library.pairself row_of_pat) patts1
         val finals = map row_of_pat patts2
         val originals = map (row_of_pat o #2) rows
         val dummy = case (originals\\finals)
                 of [] => ()
              | L => mk_functional_err
     ("The following clauses are redundant (covered by preceding clauses): " ^
                       commas (map (fn i => Int.toString (i + 1)) L))
     in {functional = Abs(Sign.base_name fname, ftype,
                          abstract_over (atom,
                                         absfree(aname,atype, case_tm))),
         pats = patts2}
    end end;

    (*----------------------------------------------------------------------------
     *
     *                    PRINCIPLES OF DEFINITION
     *
     *---------------------------------------------------------------------------*)

    (*For Isabelle, the lhs of a definition must be a constant.*)
    fun mk_const_def sign (c, Ty, rhs) =
        Sign.infer_types (Sign.pp sign) sign (K NONE) (K NONE) [] false
                   ([Const("==",dummyT) $ Const(c,Ty) $ rhs], propT)
        |> #1;

    (*Make all TVars available for instantiation by adding a ? to the front*)
    fun poly_tvars (Type(a,Ts)) = Type(a, map (poly_tvars) Ts)
      | poly_tvars (TFree (a,sort)) = TVar (("?" ^ a, 0), sort)
      | poly_tvars (TVar ((a,i),sort)) = TVar (("?" ^ a, i+1), sort);

    local val f_eq_wfrec_R_M =
               #ant(S.dest_imp(#2(S.strip_forall (concl Thms.WFREC_COROLLARY))))
          val {lhs=f, rhs} = S.dest_eq f_eq_wfrec_R_M
          val (fname,_) = dest_Free f
          val (wfrec,_) = S.strip_comb rhs
    in
    fun wfrec_definition0 thy fid R (functional as Abs(x, Ty, _)) =
     let val def_name = if x<>fid then
                            raise TFL_ERR "wfrec_definition0"
                                          ("Expected a definition of " ^
                                                 quote fid ^ " but found one of " ^
                                          quote x)
                        else x ^ "_def"
         val wfrec_R_M =  map_term_types poly_tvars
                              (wfrec $ map_term_types poly_tvars R)
                          $ functional
         val def_term = mk_const_def (Theory.sign_of thy) (x, Ty, wfrec_R_M)
         val (thy', [def]) = PureThy.add_defs_i false [Thm.no_attributes (def_name, def_term)] thy
     in (thy', def) end;
    end;

    (*---------------------------------------------------------------------------
     * This structure keeps track of congruence rules that aren't derived
     * from a datatype definition.
     *---------------------------------------------------------------------------*)
    fun extraction_thms thy =
     let val {case_rewrites,case_congs} = Thry.extract_info thy
     in (case_rewrites, case_congs)
     end;

    (*---------------------------------------------------------------------------
     * Pair patterns with termination conditions. The full list of patterns for
     * a definition is merged with the TCs arising from the user-given clauses.
     * There can be fewer clauses than the full list, if the user omitted some
     * cases. This routine is used to prepare input for mk_induction.
     *---------------------------------------------------------------------------*)
    fun merge full_pats TCs =
    let fun insert (p,TCs) =
          let fun insrt ((x as (h,[]))::rst) =
                     if (p aconv h) then (p,TCs)::rst else x::insrt rst
                | insrt (x::rst) = x::insrt rst
                | insrt[] = raise TFL_ERR "merge.insert" "pattern not found"
          in insrt end
        fun pass ([],ptcl_final) = ptcl_final
          | pass (ptcs::tcl, ptcl) = pass(tcl, insert ptcs ptcl)
    in
      pass (TCs, map (fn p => (p,[])) full_pats)
    end;

    fun givens pats = map pat_of (List.filter given pats);

    fun post_definition meta_tflCongs (theory, (def, pats)) =
     let val tych = Thry.typecheck theory
         val f = #lhs(S.dest_eq(concl def))
         val corollary = R.MATCH_MP Thms.WFREC_COROLLARY def
         val pats' = List.filter given pats
         val given_pats = map pat_of pats'
         val rows = map row_of_pat pats'
         val WFR = #ant(S.dest_imp(concl corollary))
         val R = #Rand(S.dest_comb WFR)
         val corollary' = R.UNDISCH corollary  (* put WF R on assums *)
         val corollaries = map (fn pat => R.SPEC (tych pat) corollary')
                               given_pats
         val (case_rewrites,context_congs) = extraction_thms theory
         (*case_ss causes minimal simplification: bodies of case expressions are
           not simplified. Otherwise large examples (Red-Black trees) are too
           slow.*)
         val case_ss = HOL_basic_ss addcongs
           DatatypePackage.weak_case_congs_of theory addsimps case_rewrites
         val corollaries' = map (Simplifier.simplify case_ss) corollaries
         val extract = R.CONTEXT_REWRITE_RULE
                         (f, [R], cut_apply, meta_tflCongs@context_congs)
         val (rules, TCs) = ListPair.unzip (map extract corollaries')
         val rules0 = map (rewrite_rule [Thms.CUT_DEF]) rules
         val mk_cond_rule = R.FILTER_DISCH_ALL(not o curry (op aconv) WFR)
         val rules1 = R.LIST_CONJ(map mk_cond_rule rules0)
     in
     {theory = theory,
      rules = rules1,
      rows = rows,
      full_pats_TCs = merge (map pat_of pats) (ListPair.zip (given_pats, TCs)),
      TCs = TCs}
     end;

    (*---------------------------------------------------------------------------
     * Perform the extraction without making the definition. Definition and
     * extraction commute for the non-nested case.  (Deferred recdefs)
     *
     * The purpose of wfrec_eqns is merely to instantiate the recursion theorem
     * and extract termination conditions: no definition is made.
     *---------------------------------------------------------------------------*)

    fun wfrec_eqns thy fid tflCongs eqns =
     let val {lhs,rhs} = S.dest_eq (hd eqns)
         val (f,args) = S.strip_comb lhs
         val (fname,fty) = dest_atom f
         val (SV,a) = front_last args    (* SV = schematic variables *)
         val g = list_comb(f,SV)
         val h = Free(fname,type_of g)
         val eqns1 = map (subst_free[(g,h)]) eqns
         val {functional as Abs(x, Ty, _),  pats} = mk_functional thy eqns1
         val given_pats = givens pats
         (* val f = Free(x,Ty) *)
         val Type("fun", [f_dty, f_rty]) = Ty
         val dummy = if x<>fid then
                            raise TFL_ERR "wfrec_eqns"
                                          ("Expected a definition of " ^
                                          quote fid ^ " but found one of " ^
                                          quote x)
                     else ()
         val (case_rewrites,context_congs) = extraction_thms thy
         val tych = Thry.typecheck thy
         val WFREC_THM0 = R.ISPEC (tych functional) Thms.WFREC_COROLLARY
         val Const("All",_) $ Abs(Rname,Rtype,_) = concl WFREC_THM0
         val R = Free (variant (foldr add_term_names [] eqns) Rname,
                       Rtype)
         val WFREC_THM = R.ISPECL [tych R, tych g] WFREC_THM0
         val ([proto_def, WFR],_) = S.strip_imp(concl WFREC_THM)
         val dummy =
               if !trace then
                   writeln ("ORIGINAL PROTO_DEF: " ^
                              Sign.string_of_term (Theory.sign_of thy) proto_def)
               else ()
         val R1 = S.rand WFR
         val corollary' = R.UNDISCH(R.UNDISCH WFREC_THM)
         val corollaries = map (fn pat => R.SPEC (tych pat) corollary') given_pats
         val corollaries' = map (rewrite_rule case_rewrites) corollaries
         fun extract X = R.CONTEXT_REWRITE_RULE
                           (f, R1::SV, cut_apply, tflCongs@context_congs) X
     in {proto_def = proto_def,
         SV=SV,
         WFR=WFR,
         pats=pats,
         extracta = map extract corollaries'}
     end;

    (*---------------------------------------------------------------------------
     * Define the constant after extracting the termination conditions. The
     * wellfounded relation used in the definition is computed by using the
     * choice operator on the extracted conditions (plus the condition that
     * such a relation must be wellfounded).
     *---------------------------------------------------------------------------*)

    fun lazyR_def thy fid tflCongs eqns =
     let val {proto_def,WFR,pats,extracta,SV} =
               wfrec_eqns thy fid tflCongs eqns
         val R1 = S.rand WFR
         val f = #lhs(S.dest_eq proto_def)
         val (extractants,TCl) = ListPair.unzip extracta
         val dummy = if !trace
                     then (writeln "Extractants = ";
                           prths extractants;
                           ())
                     else ()
         val TCs = foldr (gen_union (op aconv)) [] TCl
         val full_rqt = WFR::TCs
         val R' = S.mk_select{Bvar=R1, Body=S.list_mk_conj full_rqt}
         val R'abs = S.rand R'
         val proto_def' = subst_free[(R1,R')] proto_def
         val dummy = if !trace then writeln ("proto_def' = " ^
                                             Sign.string_of_term
                                             (Theory.sign_of thy) proto_def')
                               else ()
         val {lhs,rhs} = S.dest_eq proto_def'
         val (c,args) = S.strip_comb lhs
         val (name,Ty) = dest_atom c
         val defn = mk_const_def (Theory.sign_of thy)
                     (name, Ty, S.list_mk_abs (args,rhs))
         val (theory, [def0]) =
           thy
           |> PureThy.add_defs_i false
                [Thm.no_attributes (fid ^ "_def", defn)]
         val def = freezeT def0;
         val dummy = if !trace then writeln ("DEF = " ^ string_of_thm def)
                               else ()
         (* val fconst = #lhs(S.dest_eq(concl def))  *)
         val tych = Thry.typecheck theory
         val full_rqt_prop = map (Dcterm.mk_prop o tych) full_rqt
             (*lcp: a lot of object-logic inference to remove*)
         val baz = R.DISCH_ALL
                     (fold_rev R.DISCH full_rqt_prop
                      (R.LIST_CONJ extractants))
         val dum = if !trace then writeln ("baz = " ^ string_of_thm baz)
                               else ()
         val f_free = Free (fid, fastype_of f)  (*'cos f is a Const*)
         val SV' = map tych SV;
         val SVrefls = map reflexive SV'
         val def0 = (fold (fn x => fn th => R.rbeta(combination th x))
                       SVrefls def)
                    RS meta_eq_to_obj_eq
         val def' = R.MP (R.SPEC (tych R') (R.GEN (tych R1) baz)) def0
         val body_th = R.LIST_CONJ (map R.ASSUME full_rqt_prop)
         val SELECT_AX = (*in this way we hope to avoid a STATIC dependence upon
                           theory Hilbert_Choice*)
             thm "Hilbert_Choice.tfl_some"
             handle ERROR => error
        "defer_recdef requires theory Main or at least Hilbert_Choice as parent"
         val bar = R.MP (R.ISPECL[tych R'abs, tych R1] SELECT_AX) body_th
     in {theory = theory, R=R1, SV=SV,
         rules = fold (U.C R.MP) (R.CONJUNCTS bar) def',
         full_pats_TCs = merge (map pat_of pats) (ListPair.zip (givens pats, TCl)),
         patterns = pats}
     end;

    (*----------------------------------------------------------------------------
     *
     *                           INDUCTION THEOREM
     *
     *---------------------------------------------------------------------------*)

    (*------------------------  Miscellaneous function  --------------------------
     *
     *           [x_1,...,x_n]     ?v_1...v_n. M[v_1,...,v_n]
     *     -----------------------------------------------------------
     *     ( M[x_1,...,x_n], [(x_i,?v_1...v_n. M[v_1,...,v_n]),
     *                        ...
     *                        (x_j,?v_n. M[x_1,...,x_(n-1),v_n])] )
     *
     * This function is totally ad hoc. Used in the production of the induction
     * theorem. The nchotomy theorem can have clauses that look like
     *
     *     ?v1..vn. z = C vn..v1
     *
     * in which the order of quantification is not the order of occurrence of the
     * quantified variables as arguments to C. Since we have no control over this
     * aspect of the nchotomy theorem, we make the correspondence explicit by
     * pairing the incoming new variable with the term it gets beta-reduced into.
     *---------------------------------------------------------------------------*)

    fun alpha_ex_unroll (xlist, tm) =
      let val (qvars,body) = S.strip_exists tm
          val vlist = #2(S.strip_comb (S.rhs body))
          val plist = ListPair.zip (vlist, xlist)
          val args = map (the o AList.lookup (op aconv) plist) qvars
                       handle Option => sys_error
                           "TFL fault [alpha_ex_unroll]: no correspondence"
          fun build ex      []   = []
            | build (_$rex) (v::rst) =
               let val ex1 = betapply(rex, v)
               in  ex1 :: build ex1 rst
               end
         val (nex::exl) = rev (tm::build tm args)
      in
      (nex, ListPair.zip (args, rev exl))
      end;

    (*----------------------------------------------------------------------------
     *
     *             PROVING COMPLETENESS OF PATTERNS
     *
     *---------------------------------------------------------------------------*)

    fun mk_case ty_info usednames thy =
     let
     val divide = ipartition (gvvariant usednames)
     val tych = Thry.typecheck thy
     fun tych_binding(x,y) = (tych x, tych y)
     fun fail s = raise TFL_ERR "mk_case" s
     fun mk{rows=[],...} = fail"no rows"
       | mk{path=[], rows = [([], (thm, bindings))]} =
                             R.IT_EXISTS (map tych_binding bindings) thm
       | mk{path = u::rstp, rows as (p::_, _)::_} =
         let val (pat_rectangle,rights) = ListPair.unzip rows
             val col0 = map hd pat_rectangle
             val pat_rectangle' = map tl pat_rectangle
         in
         if (forall is_Free col0) (* column 0 is all variables *)
         then let val rights' = map (fn ((thm,theta),v) => (thm,theta@[(u,v)]))
                                    (ListPair.zip (rights, col0))
              in mk{path = rstp, rows = ListPair.zip (pat_rectangle', rights')}
              end
         else                     (* column 0 is all constructors *)
         let val Type (ty_name,_) = type_of p
         in
         case (ty_info ty_name)
         of NONE => fail("Not a known datatype: "^ty_name)
          | SOME{constructors,nchotomy} =>
            let val thm' = R.ISPEC (tych u) nchotomy
                val disjuncts = S.strip_disj (concl thm')
                val subproblems = divide(constructors, rows)
                val groups      = map #group subproblems
                and new_formals = map #new_formals subproblems
                val existentials = ListPair.map alpha_ex_unroll
                                       (new_formals, disjuncts)
                val constraints = map #1 existentials
                val vexl = map #2 existentials
                fun expnd tm (pats,(th,b)) = (pats,(R.SUBS[R.ASSUME(tych tm)]th,b))
                val news = map (fn (nf,rows,c) => {path = nf@rstp,
                                                   rows = map (expnd c) rows})
                               (U.zip3 new_formals groups constraints)
                val recursive_thms = map mk news
                val build_exists = Library.foldr
                                    (fn((x,t), th) =>
                                     R.CHOOSE (tych x, R.ASSUME (tych t)) th)
                val thms' = ListPair.map build_exists (vexl, recursive_thms)
                val same_concls = R.EVEN_ORS thms'
            in R.DISJ_CASESL thm' same_concls
            end
         end end
     in mk
     end;

    fun complete_cases thy =
     let val tych = Thry.typecheck thy
         val ty_info = Thry.induct_info thy
     in fn pats =>
     let val names = foldr add_term_names [] pats
         val T = type_of (hd pats)
         val aname = Term.variant names "a"
         val vname = Term.variant (aname::names) "v"
         val a = Free (aname, T)
         val v = Free (vname, T)
         val a_eq_v = HOLogic.mk_eq(a,v)
         val ex_th0 = R.EXISTS (tych (S.mk_exists{Bvar=v,Body=a_eq_v}), tych a)
                               (R.REFL (tych a))
         val th0 = R.ASSUME (tych a_eq_v)
         val rows = map (fn x => ([x], (th0,[]))) pats
     in
     R.GEN (tych a)
           (R.RIGHT_ASSOC
              (R.CHOOSE(tych v, ex_th0)
                    (mk_case ty_info (vname::aname::names)
                     thy {path=[v], rows=rows})))
     end end;

    (*---------------------------------------------------------------------------
     * Constructing induction hypotheses: one for each recursive call.
     *
     * Note. R will never occur as a variable in the ind_clause, because
     * to do so, it would have to be from a nested definition, and we don't
     * allow nested defns to have R variable.
     *
     * Note. When the context is empty, there can be no local variables.
     *---------------------------------------------------------------------------*)
    (*
    local infix 5 ==>
          fun (tm1 ==> tm2) = S.mk_imp{ant = tm1, conseq = tm2}
    in
    fun build_ih f P (pat,TCs) =
     let val globals = S.free_vars_lr pat
         fun nested tm = isSome (S.find_term (curry (op aconv) f) tm)
         fun dest_TC tm =
             let val (cntxt,R_y_pat) = S.strip_imp(#2(S.strip_forall tm))
                 val (R,y,_) = S.dest_relation R_y_pat
                 val P_y = if (nested tm) then R_y_pat ==> P$y else P$y
             in case cntxt
                  of [] => (P_y, (tm,[]))
                   | _  => let
                        val imp = S.list_mk_conj cntxt ==> P_y
                        val lvs = gen_rems (op aconv) (S.free_vars_lr imp, globals)
                        val locals = #2(U.pluck (curry (op aconv) P) lvs) handle U.ERR _ => lvs
                        in (S.list_mk_forall(locals,imp), (tm,locals)) end
             end
     in case TCs
        of [] => (S.list_mk_forall(globals, P$pat), [])
         |  _ => let val (ihs, TCs_locals) = ListPair.unzip(map dest_TC TCs)
                     val ind_clause = S.list_mk_conj ihs ==> P$pat
                 in (S.list_mk_forall(globals,ind_clause), TCs_locals)
                 end
     end
    end;
    *)

    local infix 5 ==>
          fun (tm1 ==> tm2) = S.mk_imp{ant = tm1, conseq = tm2}
    in
    fun build_ih f (P,SV) (pat,TCs) =
     let val pat_vars = S.free_vars_lr pat
         val globals = pat_vars@SV
         fun nested tm = isSome (S.find_term (curry (op aconv) f) tm)
         fun dest_TC tm =
             let val (cntxt,R_y_pat) = S.strip_imp(#2(S.strip_forall tm))
                 val (R,y,_) = S.dest_relation R_y_pat
                 val P_y = if (nested tm) then R_y_pat ==> P$y else P$y
             in case cntxt
                  of [] => (P_y, (tm,[]))
                   | _  => let
                        val imp = S.list_mk_conj cntxt ==> P_y
                        val lvs = gen_rems (op aconv) (S.free_vars_lr imp, globals)
                        val locals = #2(U.pluck (curry (op aconv) P) lvs) handle U.ERR _ => lvs
                        in (S.list_mk_forall(locals,imp), (tm,locals)) end
             end
     in case TCs
        of [] => (S.list_mk_forall(pat_vars, P$pat), [])
         |  _ => let val (ihs, TCs_locals) = ListPair.unzip(map dest_TC TCs)
                     val ind_clause = S.list_mk_conj ihs ==> P$pat
                 in (S.list_mk_forall(pat_vars,ind_clause), TCs_locals)
                 end
     end
    end;

    (*---------------------------------------------------------------------------
     * This function makes good on the promise made in "build_ih".
     *
     * Input  is tm = "(!y. R y pat ==> P y) ==> P pat",
     *           TCs = TC_1[pat] ... TC_n[pat]
     *           thm = ih1 /\ ... /\ ih_n |- ih[pat]
     *---------------------------------------------------------------------------*)
    fun prove_case f thy (tm,TCs_locals,thm) =
     let val tych = Thry.typecheck thy
         val antc = tych(#ant(S.dest_imp tm))
         val thm' = R.SPEC_ALL thm
         fun nested tm = isSome (S.find_term (curry (op aconv) f) tm)
         fun get_cntxt TC = tych(#ant(S.dest_imp(#2(S.strip_forall(concl TC)))))
         fun mk_ih ((TC,locals),th2,nested) =
             R.GENL (map tych locals)
                (if nested then R.DISCH (get_cntxt TC) th2 handle U.ERR _ => th2
                 else if S.is_imp (concl TC) then R.IMP_TRANS TC th2
                 else R.MP th2 TC)
     in
     R.DISCH antc
     (if S.is_imp(concl thm') (* recursive calls in this clause *)
      then let val th1 = R.ASSUME antc
               val TCs = map #1 TCs_locals
               val ylist = map (#2 o S.dest_relation o #2 o S.strip_imp o
                                #2 o S.strip_forall) TCs
               val TClist = map (fn(TC,lvs) => (R.SPEC_ALL(R.ASSUME(tych TC)),lvs))
                                TCs_locals
               val th2list = map (U.C R.SPEC th1 o tych) ylist
               val nlist = map nested TCs
               val triples = U.zip3 TClist th2list nlist
               val Pylist = map mk_ih triples
           in R.MP thm' (R.LIST_CONJ Pylist) end
      else thm')
     end;

    (*---------------------------------------------------------------------------
     *
     *         x = (v1,...,vn)  |- M[x]
     *    ---------------------------------------------
     *      ?v1 ... vn. x = (v1,...,vn) |- M[x]
     *
     *---------------------------------------------------------------------------*)
    fun LEFT_ABS_VSTRUCT tych thm =
      let fun CHOOSER v (tm,thm) =
            let val ex_tm = S.mk_exists{Bvar=v,Body=tm}
            in (ex_tm, R.CHOOSE(tych v, R.ASSUME (tych ex_tm)) thm)
            end
          val [veq] = List.filter (can S.dest_eq) (#1 (R.dest_thm thm))
          val {lhs,rhs} = S.dest_eq veq
          val L = S.free_vars_lr rhs
      in  #2 (fold_rev CHOOSER L (veq,thm))  end;

    (*----------------------------------------------------------------------------
     * Input : f, R,  and  [(pat1,TCs1),..., (patn,TCsn)]
     *
     * Instantiates WF_INDUCTION_THM, getting Sinduct and then tries to prove
     * recursion induction (Rinduct) by proving the antecedent of Sinduct from
     * the antecedent of Rinduct.
     *---------------------------------------------------------------------------*)
    fun mk_induction thy {fconst, R, SV, pat_TCs_list} =
    let val tych = Thry.typecheck thy
        val Sinduction = R.UNDISCH (R.ISPEC (tych R) Thms.WF_INDUCTION_THM)
        val (pats,TCsl) = ListPair.unzip pat_TCs_list
        val case_thm = complete_cases thy pats
        val domain = (type_of o hd) pats
        val Pname = Term.variant (foldr (Library.foldr add_term_names)
                                  [] (pats::TCsl)) "P"
        val P = Free(Pname, domain --> HOLogic.boolT)
        val Sinduct = R.SPEC (tych P) Sinduction
        val Sinduct_assumf = S.rand ((#ant o S.dest_imp o concl) Sinduct)
        val Rassums_TCl' = map (build_ih fconst (P,SV)) pat_TCs_list
        val (Rassums,TCl') = ListPair.unzip Rassums_TCl'
        val Rinduct_assum = R.ASSUME (tych (S.list_mk_conj Rassums))
        val cases = map (fn pat => betapply (Sinduct_assumf, pat)) pats
        val tasks = U.zip3 cases TCl' (R.CONJUNCTS Rinduct_assum)
        val proved_cases = map (prove_case fconst thy) tasks
        val v = Free (variant (foldr add_term_names [] (map concl proved_cases))
                        "v",
                      domain)
        val vtyped = tych v
        val substs = map (R.SYM o R.ASSUME o tych o (curry HOLogic.mk_eq v)) pats
        val proved_cases1 = ListPair.map (fn (th,th') => R.SUBS[th]th')
                              (substs, proved_cases)
        val abs_cases = map (LEFT_ABS_VSTRUCT tych) proved_cases1
        val dant = R.GEN vtyped (R.DISJ_CASESL (R.ISPEC vtyped case_thm) abs_cases)
        val dc = R.MP Sinduct dant
        val Parg_ty = type_of(#Bvar(S.dest_forall(concl dc)))
        val vars = map (gvvariant[Pname]) (S.strip_prod_type Parg_ty)
        val dc' = fold_rev (R.GEN o tych) vars
                           (R.SPEC (tych(S.mk_vstruct Parg_ty vars)) dc)
    in
       R.GEN (tych P) (R.DISCH (tych(concl Rinduct_assum)) dc')
    end
    handle U.ERR _ => raise TFL_ERR "mk_induction" "failed derivation";

    (*---------------------------------------------------------------------------
     *
     *                        POST PROCESSING
     *
     *---------------------------------------------------------------------------*)

    fun simplify_induction thy hth ind =
      let val tych = Thry.typecheck thy
          val (asl,_) = R.dest_thm ind
          val (_,tc_eq_tc') = R.dest_thm hth
          val tc = S.lhs tc_eq_tc'
          fun loop [] = ind
            | loop (asm::rst) =
              if (can (Thry.match_term thy asm) tc)
              then R.UNDISCH
                     (R.MATCH_MP
                         (R.MATCH_MP Thms.simp_thm (R.DISCH (tych asm) ind))
                         hth)
             else loop rst
      in loop asl
    end;

    (*---------------------------------------------------------------------------
     * The termination condition is an antecedent to the rule, and an
     * assumption to the theorem.
     *---------------------------------------------------------------------------*)
    fun elim_tc tcthm (rule,induction) =
       (R.MP rule tcthm, R.PROVE_HYP tcthm induction)

    fun trace_thms s L =
      if !trace then writeln (cat_lines (s :: map string_of_thm L))
      else ();

    fun trace_cterms s L =
      if !trace then writeln (cat_lines (s :: map string_of_cterm L))
      else ();;

    fun postprocess strict {wf_tac, terminator, simplifier} theory {rules,induction,TCs} =
    let val tych = Thry.typecheck theory
        val prove = R.prove strict;

       (*---------------------------------------------------------------------
        * Attempt to eliminate WF condition. It's the only assumption of rules
        *---------------------------------------------------------------------*)
       val (rules1,induction1)  =
           let val thm = prove(tych(HOLogic.mk_Trueprop
                                      (hd(#1(R.dest_thm rules)))),
                                 wf_tac)
           in (R.PROVE_HYP thm rules,  R.PROVE_HYP thm induction)
           end handle U.ERR _ => (rules,induction);

       (*----------------------------------------------------------------------
        * The termination condition (tc) is simplified to |- tc = tc' (there
        * might not be a change!) and then 3 attempts are made:
        *
        *   1. if |- tc = T, then eliminate it with eqT; otherwise,
        *   2. apply the terminator to tc'. If |- tc' = T then eliminate; else
        *   3. replace tc by tc' in both the rules and the induction theorem.
        *---------------------------------------------------------------------*)

       fun simplify_tc tc (r,ind) =
           let val tc1 = tych tc
               val _ = trace_cterms "TC before simplification: " [tc1]
               val tc_eq = simplifier tc1
               val _ = trace_thms "result: " [tc_eq]
           in
           elim_tc (R.MATCH_MP Thms.eqT tc_eq) (r,ind)
           handle U.ERR _ =>
            (elim_tc (R.MATCH_MP(R.MATCH_MP Thms.rev_eq_mp tc_eq)
                      (prove(tych(HOLogic.mk_Trueprop(S.rhs(concl tc_eq))),
                               terminator)))
                     (r,ind)
             handle U.ERR _ =>
              (R.UNDISCH(R.MATCH_MP (R.MATCH_MP Thms.simp_thm r) tc_eq),
               simplify_induction theory tc_eq ind))
           end

       (*----------------------------------------------------------------------
        * Nested termination conditions are harder to get at, since they are
        * left embedded in the body of the function (and in induction
        * theorem hypotheses). Our "solution" is to simplify them, and try to
        * prove termination, but leave the application of the resulting theorem
        * to a higher level. So things go much as in "simplify_tc": the
        * termination condition (tc) is simplified to |- tc = tc' (there might
        * not be a change) and then 2 attempts are made:
        *
        *   1. if |- tc = T, then return |- tc; otherwise,
        *   2. apply the terminator to tc'. If |- tc' = T then return |- tc; else
        *   3. return |- tc = tc'
        *---------------------------------------------------------------------*)
       fun simplify_nested_tc tc =
          let val tc_eq = simplifier (tych (#2 (S.strip_forall tc)))
          in
          R.GEN_ALL
           (R.MATCH_MP Thms.eqT tc_eq
            handle U.ERR _ =>
              (R.MATCH_MP(R.MATCH_MP Thms.rev_eq_mp tc_eq)
                          (prove(tych(HOLogic.mk_Trueprop (S.rhs(concl tc_eq))),
                                   terminator))
                handle U.ERR _ => tc_eq))
          end

       (*-------------------------------------------------------------------
        * Attempt to simplify the termination conditions in each rule and
        * in the induction theorem.
        *-------------------------------------------------------------------*)
       fun strip_imp tm = if S.is_neg tm then ([],tm) else S.strip_imp tm
       fun loop ([],extras,R,ind) = (rev R, ind, extras)
         | loop ((r,ftcs)::rst, nthms, R, ind) =
            let val tcs = #1(strip_imp (concl r))
                val extra_tcs = gen_rems (op aconv) (ftcs, tcs)
                val extra_tc_thms = map simplify_nested_tc extra_tcs
                val (r1,ind1) = fold simplify_tc tcs (r,ind)
                val r2 = R.FILTER_DISCH_ALL(not o S.is_WFR) r1
            in loop(rst, nthms@extra_tc_thms, r2::R, ind1)
            end
       val rules_tcs = ListPair.zip (R.CONJUNCTS rules1, TCs)
       val (rules2,ind2,extras) = loop(rules_tcs,[],[],induction1)
    in
      {induction = ind2, rules = R.LIST_CONJ rules2, nested_tcs = extras}
    end;

    end;