TFL/tfl.sml
 author wenzelm Thu, 11 Mar 1999 12:33:34 +0100 changeset 6347 ce7ab97a8e15 parent 6092 d9db67970c73 child 6397 e70ae9b575cc permissions -rw-r--r--
tuned;
```
(*  Title:      TFL/tfl
ID:         \$Id\$
Author:     Konrad Slind, Cambridge University Computer Laboratory

Main module
*)

structure Prim : TFL_sig =
struct

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

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 l =
rev(#1(foldl (fn ((alist,i), x) => ((x,i)::alist, i+1)) (([],0), l)));

fun stringize [] = ""
| stringize [i] = Int.toString i
| stringize (h::t) = (Int.toString 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.
*
* 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 := 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{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) = dest_Const c
val L = binder_types 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 (#1(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 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 term * int
| OMITTED of term * int

fun pattern_map f (GIVEN (tm,i)) = GIVEN(f tm, i)
| pattern_map f (OMITTED (tm,i)) = OMITTED(f tm, i);

fun pattern_subst theta = pattern_map (subst_free theta);

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) = 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 =
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 ((#1(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) = 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,l) =
let val L = length (binder_types (type_of c))
fun build (prefix,tag,plist) =
let val args   = take (L,plist)
and plist' = drop(L,plist)
in (prefix,tag,list_comb(c,args)::plist') end
in map build l 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 usednames range_ty =
let
fun mk_case_fail s = raise TFL_ERR{func = "mk_case", mesg = 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 ((prefix, 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 ((prefix, 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 = ((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(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_prefix 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 open BasisLibrary (*restore original List*)
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,Ty} => [Free(Name,Ty)]
| 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 mem_term (v,rst) then
raise TFL_ERR{func = "no_repeat_vars",
mesg = 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;

local fun mk_functional_err s = raise TFL_ERR{func = "mk_functional", mesg=s}
fun single [f] = f
| single fs  = mk_functional_err (Int.toString (length fs) ^
" distinct function names!")
in
fun mk_functional thy clauses =
let val (L,R) = ListPair.unzip
(map (fn (Const("op =",_) \$ t \$ u) => (t,u)) clauses)
val (funcs,pats) = ListPair.unzip (map (fn (t\$u) =>(t,u)) L)
val fcon as Const (fname, ftype) = single (gen_distinct (op aconv) funcs)
val dummy = map (no_repeat_vars thy) pats
val rows = ListPair.zip (map (fn x => ([],[x])) pats,
map GIVEN (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,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
val dummy = case (originals\\finals)
of [] => ()
| L => mk_functional_err("The following rows (counting from zero)\
\ are inaccessible: "^stringize L)
in {functional = Abs(Sign.base_name fname, ftype,
abstract_over (fcon,
absfree(aname,atype, case_tm))),
pats = patts2}
end end;

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

(*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(Name, Ty, _)) =
let val def_name = if Name<>fid then
raise TFL_ERR{func = "wfrec_definition0",
mesg = "Expected a definition of " ^
quote fid ^ " but found one of " ^
quote Name}
else Name ^ "_def"
val wfrec_R_M =  map_term_types poly_tvars
(wfrec \$ map_term_types poly_tvars R)
\$ functional
val (def_term, _) =
Sign.infer_types (sign_of thy) (K None) (K None) [] false
([Const("==",dummyT) \$ Const(Name,Ty) \$ wfrec_R_M],
propT)
in  PureThy.add_defs_i [Thm.no_attributes (def_name, def_term)] thy  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{func="merge.insert",
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 [] = []
| givens (GIVEN(tm,_)::pats) = tm :: givens pats
| givens (OMITTED _::pats)   = givens pats;

fun post_definition (ss, 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 given_pats = givens 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
val corollaries' = map(rewrite_rule case_rewrites) corollaries
val extract = R.CONTEXT_REWRITE_RULE
(ss, f, R,
R.ISPECL (map tych [f,R]) Thms.CUT_LEMMA,
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,   (* holds def, if it's needed *)
rules = rules1,
full_pats_TCs = merge (map pat_of pats)
(ListPair.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.
* CURRENTLY UNUSED
fun wfrec_eqns (ss, tflCongs) 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, Type("fun", [f_dty, f_rty])) = dest_Free f
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 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 (fn pat => R.SPEC (tych pat) corollary') given_pats
val corollaries' = map (rewrite_rule case_rewrites) corollaries
val extract = R.CONTEXT_REWRITE_RULE
(ss, f, R1,
R.ISPECL (map tych [f,R1]) Thms.CUT_LEMMA,
tflCongs@context_congs)
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).
* CURRENTLY UNUSED
fun lazyR_def ss thy eqns =
let val {proto_def,WFR,pats,extracta} = wfrec_eqns ss thy eqns
val R1 = S.rand WFR
val f = S.lhs proto_def
val (Name,_) = dest_Free f
val (extractants,TCl) = ListPair.unzip extracta
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 theory =
thy
|> PureThy.add_defs_i [Thm.no_attributes (Name ^ "_def", subst_free[(R1,R')] proto_def)];
val def = freezeT((get_axiom theory (Name ^ "_def")) RS meta_eq_to_obj_eq)
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.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) (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 (fn qv => the (gen_assoc (op aconv) (plist, qv))) qvars
handle OPTION => 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{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) = 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 = 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 = is_some (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 _ => 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;

(*---------------------------------------------------------------------------
* 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 = is_some (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 _ => 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] = 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  #2 (U.itlist 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 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) = ListPair.unzip pat_TCs_list
val case_thm = complete_cases thy pats
val domain = (type_of o hd) pats
val Pname = Term.variant (foldr (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 f P) 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 f 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' = 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(HOLogic.mk_Trueprop
(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(HOLogic.mk_Trueprop(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(HOLogic.mk_Trueprop (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 = gen_rems (op 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 = 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; (* TFL *)
```