TFL/tfl.sml

added header

functor TFL(structure Rules : Rules_sig
            structure Thry  : Thry_sig
            structure Thms  : Thms_sig
            sharing type Rules.binding = Thry.binding = 
                         Thry.USyntax.binding = Mask.binding
            sharing type Rules.Type = Thry.Type = Thry.USyntax.Type
            sharing type Rules.Preterm = Thry.Preterm = Thry.USyntax.Preterm
            sharing type Rules.Term = Thry.Term = Thry.USyntax.Term
            sharing type Thms.Thm = Rules.Thm = Thry.Thm) : TFL_sig  =
struct

(* Declarations *)
structure Thms = Thms;
structure Rules = Rules;
structure Thry = Thry;
structure USyntax = Thry.USyntax;

type Preterm = Thry.USyntax.Preterm;
type Term = Thry.USyntax.Term;
type Thm = Thms.Thm;
type Thry = Thry.Thry;
type Tactic = Rules.Tactic;
   

(* Abbreviations *)
structure R = Rules;
structure S = USyntax;
structure U = S.Utils;

(* Declares 'a binding datatype *)
open Mask;

nonfix mem --> |-> ##;
val --> = S.-->;
val ##    = U.##;

infixr 3 -->;
infixr 7 |->;
infix  4 ##; 

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

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

fun flatten [] = []
  | flatten (h::t) = h@flatten t;


fun gtake f =
  let fun grab(0,rst) = ([],rst)
        | grab(n, x::rst) = 
             let val (taken,left) = grab(n-1,rst)
             in (f x::taken, left) end
  in grab
  end;

fun enumerate L = 
 rev(#1(U.rev_itlist (fn x => fn (alist,i) => ((x,i)::alist, i+1)) L ([],0)));

fun stringize [] = ""
  | stringize [i] = U.int_to_string i
  | stringize (h::t) = (U.int_to_string h^", "^stringize t);


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


(*---------------------------------------------------------------------------
 * The next function is common to pattern-match translation and 
 * proof of completeness of cases for the induction theorem.
 *
 * "gvvariant" make variables that are guaranteed not to be in vlist and
 * furthermore, are guaranteed not to be equal to each other. The names of
 * the variables will start with "v" and end in a number.
 *---------------------------------------------------------------------------*)
local val counter = ref 0
in
fun gvvariant vlist =
  let val slist = ref (map (#Name o S.dest_var) vlist)
      val mem = U.mem (U.curry (op=))
      val _ = counter := 0
      fun pass str = 
         if (mem str (!slist)) 
         then ( counter := !counter + 1;
                pass (U.concat"v" (U.int_to_string(!counter))))
         else (slist := str :: !slist; str)
  in 
  fn ty => S.mk_var{Name=pass "v",  Ty=ty}
  end
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{func = "partition.part", mesg = 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 {Name,Ty} = S.dest_const c
              val (L,_) = S.strip_type Ty
              val (in_group, not_in_group) =
               U.itlist (fn (row as (p::rst, rhs)) =>
                         fn (in_group,not_in_group) =>
                  let val (pc,args) = S.strip_comb p
                  in if (#Name(S.dest_const pc) = Name)
                     then ((args@rst, rhs)::in_group, not_in_group)
                     else (in_group, row::not_in_group)
                  end)      rows ([],[])
              val col_types = U.take S.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;



(*---------------------------------------------------------------------------
 * This datatype carries some information about the origin of a
 * clause in a function definition.
 *---------------------------------------------------------------------------*)
datatype pattern = GIVEN   of S.Preterm * int
                 | OMITTED of S.Preterm * int

fun psubst theta (GIVEN (tm,i)) = GIVEN(S.subst theta tm, i)
  | psubst theta (OMITTED (tm,i)) = OMITTED(S.subst theta tm, i);

fun dest_pattern (GIVEN (tm,i)) = ((GIVEN,i),tm)
  | dest_pattern (OMITTED (tm,i)) = ((OMITTED,i),tm);

val pat_of = #2 o dest_pattern;
val row_of_pat = #2 o #1 o dest_pattern;

(*---------------------------------------------------------------------------
 * Produce an instance of a constructor, plus genvars for its arguments.
 *---------------------------------------------------------------------------*)
fun fresh_constr ty_match colty gv c =
  let val {Ty,...} = S.dest_const c
      val (L,ty) = S.strip_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 =
  U.itlist (fn (row as ((prefix, p::rst), rhs)) =>
            fn (in_group,not_in_group) =>
               let val (pc,args) = S.strip_comb p
               in if ((#Name(S.dest_const pc) = Name) handle _ => false)
                  then (((prefix,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{func="partition", mesg="no rows"}
  | partition gv ty_match
              (constructors, colty, res_ty, rows as (((prefix,_),_)::_)) =
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 {Name,Ty} = S.dest_const c'
             val (in_group, not_in_group) = mk_group Name rows
             val in_group' =
                 if (null in_group)  (* Constructor not given *)
                 then [((prefix, #2(fresh c)), OMITTED (S.ARB res_ty, ~1))]
                 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 =
  let val L = length(#1(S.strip_type(S.type_of c)))
      fun build (prefix,tag,plist) =
          let val (args,plist') = gtake U.I (L, plist)
           in (prefix,tag,S.list_mk_comb(c,args)::plist') end
  in map build 
  end;

fun v_to_prefix (prefix, v::pats) = (v::prefix,pats)
  | v_to_prefix _ = raise TFL_ERR{func="mk_case", mesg="v_to_prefix"};

fun v_to_pats (v::prefix,tag, pats) = (prefix, tag, v::pats)
  | v_to_pats _ = raise TFL_ERR{func="mk_case", mesg="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 FV range_ty =
 let 
 fun mk_case_fail s = raise TFL_ERR{func = "mk_case", mesg = s}
 val fresh_var = gvvariant FV 
 val divide = partition fresh_var ty_match
 fun expand constructors ty ((_,[]), _) = mk_case_fail"expand_var_row"
   | expand constructors ty (row as ((prefix, p::rst), rhs)) = 
       if (S.is_var p) 
       then let val fresh = fresh_constr ty_match ty fresh_var
                fun expnd (c,gvs) = 
                  let val capp = S.list_mk_comb(c,gvs)
                  in ((prefix, capp::rst), psubst[p |-> capp] rhs)
                  end
            in map expnd (map fresh constructors)  end
       else [row]
 fun mk{rows=[],...} = mk_case_fail"no rows"
   | mk{path=[], rows = ((prefix, []), rhs)::_} =  (* Done *)
        let val (tag,tm) = dest_pattern rhs
        in ([(prefix,tag,[])], tm)
        end
   | mk{path=[], rows = _::_} = mk_case_fail"blunder"
   | mk{path as u::rstp, rows as ((prefix, []), rhs)::rst} = 
        mk{path = path, 
           rows = ((prefix, [fresh_var(S.type_of u)]), rhs)::rst}
   | mk{path = u::rstp, rows as ((_, p::_), _)::_} =
     let val (pat_rectangle,rights) = U.unzip rows
         val col0 = map(hd o #2) pat_rectangle
     in 
     if (U.all S.is_var col0) 
     then let val rights' = map(fn(v,e) => psubst[v|->u] e) (U.zip col0 rights)
              val pat_rectangle' = map v_to_prefix pat_rectangle
              val (pref_patl,tm) = mk{path = rstp,
                                      rows = U.zip pat_rectangle' rights'}
          in (map v_to_pats pref_patl, tm)
          end
     else
     let val pty = S.type_of p
         val ty_name = (#Tyop o S.dest_type) pty
     in
     case (ty_info ty_name)
     of U.NONE => mk_case_fail("Not a known datatype: "^ty_name)
      | U.SOME{case_const,constructors} =>
        let val case_const_name = #Name(S.dest_const case_const)
            val nrows = flatten (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})
                           (U.zip new_formals groups)
            val rec_calls = map mk news
            val (pat_rect,dtrees) = U.unzip rec_calls
            val case_functions = map S.list_mk_abs(U.zip new_formals dtrees)
            val types = map S.type_of (case_functions@[u]) @ [range_ty]
            val case_const' = S.mk_const{Name = case_const_name,
                                         Ty   = list_mk_type types}
            val tree = S.list_mk_comb(case_const', case_functions@[u])
            val pat_rect1 = flatten(U.map2 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 v => [S.mk_var v]
      | S.CONST _ => []
      | S.COMB{Rator, Rand} => FV_multiset Rator @ FV_multiset Rand
      | S.LAMB _ => raise TFL_ERR{func = "FV_multiset", mesg = "lambda"};

fun no_repeat_vars thy pat =
 let fun check [] = true
       | check (v::rst) =
         if (U.mem S.aconv v rst) 
         then raise TFL_ERR{func = "no_repeat_vars",
             mesg = U.concat(U.quote(#Name(S.dest_var v)))
                     (U.concat" occurs repeatedly in the pattern "
                         (U.quote(S.Term_to_string (Thry.typecheck thy pat))))}
         else check rst
 in check (FV_multiset pat)
 end;

local fun paired1{lhs,rhs} = (lhs,rhs) 
      and paired2{Rator,Rand} = (Rator,Rand)
      fun mk_functional_err s = raise TFL_ERR{func = "mk_functional", mesg=s}
in
fun mk_functional thy eqs =
 let val clauses = S.strip_conj eqs
     val (L,R) = U.unzip (map (paired1 o S.dest_eq o U.snd o S.strip_forall)
                              clauses)
     val (funcs,pats) = U.unzip(map (paired2 o S.dest_comb) L)
     val [f] = U.mk_set (S.aconv) funcs 
               handle _ => mk_functional_err "function name not unique"
     val _ = map (no_repeat_vars thy) pats
     val rows = U.zip (map (fn x => ([],[x])) pats) (map GIVEN (enumerate R))
     val fvs = S.free_varsl R
     val a = S.variant fvs (S.mk_var{Name="a", Ty = S.type_of(hd pats)})
     val FV = a::fvs
     val ty_info = Thry.match_info thy
     val ty_match = Thry.match_type thy
     val range_ty = S.type_of (hd R)
     val (patts, case_tm) = mk_case ty_info ty_match FV range_ty 
                                    {path=[a], rows=rows}
     val patts1 = map (fn (_,(tag,i),[pat]) => tag (pat,i)) patts handle _ 
                  => mk_functional_err "error in pattern-match translation"
     val patts2 = U.sort(fn p1=>fn p2=> row_of_pat p1 < row_of_pat p2) patts1
     val finals = map row_of_pat patts2
     val originals = map (row_of_pat o #2) rows
     fun int_eq i1 (i2:int) =  (i1=i2)
     val _ = case (U.set_diff int_eq originals finals)
             of [] => ()
          | L => mk_functional_err("The following rows (counting from zero)\
                                   \ are inaccessible: "^stringize L)
 in {functional = S.list_mk_abs ([f,a], case_tm),
     pats = patts2}
end end;


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


(*----------------------------------------------------------------------------
 * This basic principle of definition takes a functional M and a relation R
 * and specializes the following theorem
 *
 *    |- !M R f. (f = WFREC R M) ==> WF R ==> !x. f x = M (f%R,x) x
 *
 * to them (getting "th1", say). Then we make the definition "f = WFREC R M" 
 * and instantiate "th1" to the constant "f" (getting th2). Then we use the
 * definition to delete the first antecedent to th2. Hence the result in
 * the "corollary" field is 
 *
 *    |-  WF R ==> !x. f x = M (f%R,x) x
 *
 *---------------------------------------------------------------------------*)

fun prim_wfrec_definition thy {R, functional} =
 let val tych = Thry.typecheck thy
     val {Bvar,...} = S.dest_abs functional
     val {Name,...} = S.dest_var Bvar  (* Intended name of definition *)
     val cor1 = R.ISPEC (tych functional) Thms.WFREC_COROLLARY
     val cor2 = R.ISPEC (tych R) cor1
     val f_eq_WFREC_R_M = (#ant o S.dest_imp o #Body 
                           o S.dest_forall o concl) cor2
     val {lhs,rhs} = S.dest_eq f_eq_WFREC_R_M
     val {Ty, ...} = S.dest_var lhs
     val def_term = S.mk_eq{lhs = S.mk_var{Name=Name,Ty=Ty}, rhs=rhs}
     val (def_thm,thy1) = Thry.make_definition thy 
                                  (U.concat Name "_def") def_term
     val (_,[f,_]) = (S.strip_comb o concl) def_thm
     val cor3 = R.ISPEC (Thry.typecheck thy1 f) cor2
 in 
 {theory = thy1, def=def_thm, corollary=R.MP cor3 def_thm}
 end;


(*---------------------------------------------------------------------------
 * This structure keeps track of congruence rules that aren't derived
 * from a datatype definition.
 *---------------------------------------------------------------------------*)
structure Context =
struct
  val non_datatype_context = ref []:Rules.Thm list ref
  fun read() = !non_datatype_context
  fun write L = (non_datatype_context := L)
end;

fun extraction_thms thy = 
 let val {case_rewrites,case_congs} = Thry.extract_info thy
 in (case_rewrites, [e-mail]())
 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 (S.aconv p h) then (p,TCs)::rst else x::insrt rst
            | insrt (x::rst) = x::insrt rst
            | insrt[] = raise TFL_ERR{func="merge.insert",mesg="pat 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 not_omitted (GIVEN(tm,_)) = tm
  | not_omitted (OMITTED _) = raise TFL_ERR{func="not_omitted",mesg=""}
val givens = U.mapfilter not_omitted;


(*--------------------------------------------------------------------------
 * This is a wrapper for "prim_wfrec_definition": it builds a functional,
 * calls "prim_wfrec_definition", then specializes the result. This gives a
 * list of rewrite rules where the right hand sides are quite ugly, so we 
 * simplify to get rid of the case statements. In essence, this function
 * performs pre- and post-processing for patterns. As well, after
 * simplification, termination conditions are extracted.
 *-------------------------------------------------------------------------*)

fun gen_wfrec_definition thy {R, eqs} =
 let val {functional,pats}  = mk_functional thy eqs
     val given_pats = givens pats
     val {def,corollary,theory} = prim_wfrec_definition thy
                                        {R=R, functional=functional}
     val tych = Thry.typecheck theory 
     val {lhs=f,...} = S.dest_eq(concl def)
     val WFR = #ant(S.dest_imp(concl corollary))
     val corollary' = R.UNDISCH corollary  (* put WF R on assums *)
     val corollaries = map (U.C R.SPEC corollary' o tych) given_pats
     val (case_rewrites,context_congs) = extraction_thms thy
     val corollaries' = map(R.simplify case_rewrites) corollaries
     fun xtract th = R.CONTEXT_REWRITE_RULE(f,R)
                         {thms = [(R.ISPECL o map tych)[f,R] Thms.CUT_LEMMA],
                         congs = context_congs,
                            th = th}
     val (rules, TCs) = U.unzip (map xtract corollaries')
     val rules0 = map (R.simplify [Thms.CUT_DEF]) rules
     val mk_cond_rule = R.FILTER_DISCH_ALL(not o S.aconv WFR)
     val rules1 = R.LIST_CONJ(map mk_cond_rule rules0)
 in
 {theory = theory,   (* holds def, if it's needed *)
  rules = rules1,
  full_pats_TCs = merge (map pat_of pats) (U.zip given_pats TCs), 
  TCs = TCs, 
  patterns = pats}
 end;


(*---------------------------------------------------------------------------
 * Perform the extraction without making the definition. Definition and
 * extraction commute for the non-nested case. For hol90 users, this 
 * function can be invoked without being in draft mode.
 *---------------------------------------------------------------------------*)
fun wfrec_eqns thy eqns =
 let val {functional,pats} = mk_functional thy eqns
     val given_pats = givens pats
     val {Bvar = f, Body} = S.dest_abs functional
     val {Bvar = x, ...} = S.dest_abs Body
     val {Name,Ty = fty} = S.dest_var f
     val {Tyop="fun", Args = [f_dty, f_rty]} = S.dest_type fty
     val (case_rewrites,context_congs) = extraction_thms thy
     val tych = Thry.typecheck thy
     val WFREC_THM0 = R.ISPEC (tych functional) Thms.WFREC_COROLLARY
     val R = S.variant(S.free_vars eqns) 
                      (#Bvar(S.dest_forall(concl WFREC_THM0)))
     val WFREC_THM = R.ISPECL [tych R, tych f] WFREC_THM0
     val ([proto_def, WFR],_) = S.strip_imp(concl WFREC_THM)
     val R1 = S.rand WFR
     val corollary' = R.UNDISCH(R.UNDISCH WFREC_THM)
     val corollaries = map (U.C R.SPEC corollary' o tych) given_pats
     val corollaries' = map (R.simplify case_rewrites) corollaries
     fun extract th = R.CONTEXT_REWRITE_RULE(f,R1)
                        {thms = [(R.ISPECL o map tych)[f,R1] Thms.CUT_LEMMA], 
                        congs = context_congs,
                          th  = th}
 in {proto_def=proto_def, 
     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 eqns =
 let val {proto_def,WFR,pats,extracta} = wfrec_eqns thy eqns
     val R1 = S.rand WFR
     val f = S.lhs proto_def
     val {Name,...} = S.dest_var f
     val (extractants,TCl) = U.unzip extracta
     val TCs = U.Union S.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 (def,theory) = Thry.make_definition thy (U.concat Name "_def") 
                                                 (S.subst[R1 |-> R'] proto_def)
     val fconst = #lhs(S.dest_eq(concl def)) 
     val tych = Thry.typecheck theory
     val baz = R.DISCH (tych proto_def)
                 (U.itlist (R.DISCH o tych) full_rqt (R.LIST_CONJ extractants))
     val def' = R.MP (R.SPEC (tych fconst) 
                             (R.SPEC (tych R') (R.GENL[tych R1, tych f] baz)))
                     def
     val body_th = R.LIST_CONJ (map (R.ASSUME o tych) full_rqt)
     val bar = R.MP (R.BETA_RULE(R.ISPECL[tych R'abs, tych R1] Thms.SELECT_AX))
                     body_th
 in {theory = theory, R=R1,
     rules = U.rev_itlist (U.C R.MP) (R.CONJUNCTS bar) def',
     full_pats_TCs = merge (map pat_of pats) (U.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 = U.zip vlist xlist
      val args = map (U.C (U.assoc1 (U.uncurry S.aconv)) plist) qvars
      val args' = map (fn U.SOME(_,v) => v 
                        | U.NONE => raise TFL_ERR{func = "alpha_ex_unroll",
                                       mesg = "no correspondence"}) args
      fun build ex [] = []
        | build ex (v::rst) =
           let val ex1 = S.beta_conv(S.mk_comb{Rator=S.rand ex, Rand=v})
           in ex1::build ex1 rst
           end
     val (nex::exl) = rev (tm::build tm args')
  in 
  (nex, U.zip args' (rev exl))
  end;



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

fun mk_case ty_info FV thy =
 let 
 val divide = ipartition (gvvariant FV)
 val tych = Thry.typecheck thy
 fun tych_binding(x|->y) = (tych x |-> tych y)
 fun fail s = raise TFL_ERR{func = "mk_case", mesg = 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) = U.unzip rows
         val col0 = map hd pat_rectangle
         val pat_rectangle' = map tl pat_rectangle
     in 
     if (U.all S.is_var col0) (* column 0 is all variables *)
     then let val rights' = map (fn ((thm,theta),v) => (thm,theta@[u|->v]))
                                (U.zip rights col0)
          in mk{path = rstp, rows = U.zip pat_rectangle' rights'}
          end
     else                     (* column 0 is all constructors *)
     let val ty_name = (#Tyop o S.dest_type o S.type_of) p
     in
     case (ty_info ty_name)
     of U.NONE => fail("Not a known datatype: "^ty_name)
      | U.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 = U.map2 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 = U.itlist(R.CHOOSE o (tych##(R.ASSUME o tych)))
            val thms' = U.map2 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
     fun pmk_var n ty = S.mk_var{Name = n,Ty = ty}
     val ty_info = Thry.induct_info thy
 in fn pats =>
 let val FV0 = S.free_varsl pats
     val a = S.variant FV0 (pmk_var "a" (S.type_of(hd pats)))
     val v = S.variant (a::FV0) (pmk_var "v" (S.type_of a))
     val FV = a::v::FV0
     val a_eq_v = S.mk_eq{lhs = a, rhs = v}
     val ex_th0 = R.EXISTS ((tych##tych) (S.mk_exists{Bvar=v,Body=a_eq_v},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 FV 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 nonfix ^ ;   infix 9 ^  ;     infix 5 ==>
      fun (tm1 ^ tm2)   = S.mk_comb{Rator = tm1, Rand = tm2}
      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 = U.can(S.find_term (S.aconv f)) tm handle _ => false
     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 = U.set_diff S.aconv (S.free_vars_lr imp) globals
                    val locals = #2(U.pluck (S.aconv P) lvs) handle _ => 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) = U.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;



(*---------------------------------------------------------------------------
 * This function makes good on the promise made in "build_ih: we prove
 * <something>.
 *
 * 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 = U.can(S.find_term (S.aconv f)) tm handle _ => false
     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 _ => 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] = U.filter (U.can S.dest_eq) (#1 (R.dest_thm thm))
      val {lhs,rhs} = S.dest_eq veq
      val L = S.free_vars_lr rhs
  in U.snd(U.itlist CHOOSER L (veq,thm))
  end;


fun combize M N = S.mk_comb{Rator=M,Rand=N};
fun eq v tm = S.mk_eq{lhs=v,rhs=tm};


(*----------------------------------------------------------------------------
 * 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 f R pat_TCs_list =
let val tych = Thry.typecheck thy
    val Sinduction = R.UNDISCH (R.ISPEC (tych R) Thms.WF_INDUCTION_THM)
    val (pats,TCsl) = U.unzip pat_TCs_list
    val case_thm = complete_cases thy pats
    val domain = (S.type_of o hd) pats
    val P = S.variant (S.all_varsl (pats@flatten TCsl))
                      (S.mk_var{Name="P", Ty=domain --> S.bool})
    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 f P) pat_TCs_list
    val (Rassums,TCl') = U.unzip Rassums_TCl'
    val Rinduct_assum = R.ASSUME (tych (S.list_mk_conj Rassums))
    val cases = map (S.beta_conv o combize Sinduct_assumf) pats
    val tasks = U.zip3 cases TCl' (R.CONJUNCTS Rinduct_assum)
    val proved_cases = map (prove_case f thy) tasks
    val v = S.variant (S.free_varsl (map concl proved_cases))
                      (S.mk_var{Name="v", Ty=domain})
    val vtyped = tych v
    val substs = map (R.SYM o R.ASSUME o tych o eq v) pats
    val proved_cases1 = U.map2 (fn th => R.SUBS[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 = S.type_of(#Bvar(S.dest_forall(concl dc)))
    val vars = map (gvvariant[P]) (S.strip_prod_type Parg_ty)
    val dc' = U.itlist (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 _ => raise TFL_ERR{func = "mk_induction", mesg = "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 (U.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 postprocess{WFtac, terminator, simplifier} theory {rules,induction,TCs} =
let val tych = Thry.typecheck theory

   (*---------------------------------------------------------------------
    * Attempt to eliminate WF condition. It's the only assumption of rules
    *---------------------------------------------------------------------*)
   val (rules1,induction1)  = 
       let val thm = R.prove(tych(hd(#1(R.dest_thm rules))),WFtac)
       in (R.PROVE_HYP thm rules,  R.PROVE_HYP thm induction)
       end handle _ => (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 tc_eq = simplifier (tych tc)
       in 
       elim_tc (R.MATCH_MP Thms.eqT tc_eq) (r,ind)
       handle _ => 
        (elim_tc (R.MATCH_MP(R.MATCH_MP Thms.rev_eq_mp tc_eq)
                            (R.prove(tych(S.rhs(concl tc_eq)),terminator)))
                 (r,ind)
         handle _ => 
          (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 _
        => (R.MATCH_MP(R.MATCH_MP Thms.rev_eq_mp tc_eq)
                      (R.prove(tych(S.rhs(concl tc_eq)),terminator))
            handle _ => 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 = U.set_diff S.aconv ftcs tcs
            val extra_tc_thms = map simplify_nested_tc extra_tcs
            val (r1,ind1) = U.rev_itlist 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 = U.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;


(*---------------------------------------------------------------------------
 * Extract termination goals so that they can be put it into a goalstack, or 
 * have a tactic directly applied to them.
 *--------------------------------------------------------------------------*)
local exception IS_NEG 
      fun strip_imp tm = if S.is_neg tm then raise IS_NEG else S.strip_imp tm
in
fun termination_goals rules = 
    U.itlist (fn th => fn A =>
        let val tcl = (#1 o S.strip_imp o #2 o S.strip_forall o concl) th
        in tcl@A
        end handle _ => A) (R.CONJUNCTS rules) (hyp rules)
end;

end; (* TFL *)