3 Author: Konrad Slind, Cambridge University Computer Laboratory
4 Copyright 1997 University of Cambridge
6 First part of main module.
13 val mk_functional: theory -> term list -> {functional: term, pats: pattern list}
14 val wfrec_definition0: theory -> string -> term -> term -> theory * thm
15 val post_definition: thm list -> theory * (thm * pattern list) ->
20 full_pats_TCs: (term * term list) list}
21 val wfrec_eqns: theory -> xstring -> thm list -> term list ->
25 extracta: (thm * term list) list,
27 val lazyR_def: theory -> xstring -> thm list -> term list ->
32 full_pats_TCs: (term * term list) list,
33 patterns : pattern list}
34 val mk_induction: theory ->
35 {fconst: term, R: term, SV: term list, pat_TCs_list: (term * term list) list} -> thm
36 val postprocess: bool -> {wf_tac: tactic, terminator: tactic, simplifier: cterm -> thm}
37 -> theory -> {rules: thm, induction: thm, TCs: term list list}
38 -> {rules: thm, induction: thm, nested_tcs: thm list}
41 structure Prim: PRIM =
44 val trace = ref false;
49 structure S = USyntax;
53 fun TFL_ERR func mesg = U.ERR {module = "Tfl", func = func, mesg = mesg};
55 val concl = #2 o R.dest_thm;
56 val hyp = #1 o R.dest_thm;
58 val list_mk_type = U.end_itlist (curry (op -->));
60 fun enumerate xs = ListPair.zip(xs, 0 upto (length xs - 1));
62 fun front_last [] = raise TFL_ERR "front_last" "empty list"
63 | front_last [x] = ([],x)
65 let val (pref,x) = front_last t
71 (*---------------------------------------------------------------------------
72 * The next function is common to pattern-match translation and
73 * proof of completeness of cases for the induction theorem.
75 * The curried function "gvvariant" returns a function to generate distinct
76 * variables that are guaranteed not to be in names. The names of
77 * the variables go u, v, ..., z, aa, ..., az, ... The returned
78 * function contains embedded refs!
79 *---------------------------------------------------------------------------*)
81 let val slist = ref names
84 if !vname mem_string (!slist)
85 then (vname := Symbol.bump_string (!vname); new())
86 else (slist := !vname :: !slist; !vname)
88 fn ty => Free(new(), ty)
92 (*---------------------------------------------------------------------------
93 * Used in induction theorem production. This is the simple case of
94 * partitioning up pattern rows by the leading constructor.
95 *---------------------------------------------------------------------------*)
96 fun ipartition gv (constructors,rows) =
97 let fun pfail s = raise TFL_ERR "partition.part" s
98 fun part {constrs = [], rows = [], A} = rev A
99 | part {constrs = [], rows = _::_, A} = pfail"extra cases in defn"
100 | part {constrs = _::_, rows = [], A} = pfail"cases missing in defn"
101 | part {constrs = c::crst, rows, A} =
102 let val (Name,Ty) = dest_Const c
103 val L = binder_types Ty
104 val (in_group, not_in_group) =
105 U.itlist (fn (row as (p::rst, rhs)) =>
106 fn (in_group,not_in_group) =>
107 let val (pc,args) = S.strip_comb p
108 in if (#1(dest_Const pc) = Name)
109 then ((args@rst, rhs)::in_group, not_in_group)
110 else (in_group, row::not_in_group)
112 val col_types = U.take type_of (length L, #1(hd in_group))
114 part{constrs = crst, rows = not_in_group,
115 A = {constructor = c,
116 new_formals = map gv col_types,
117 group = in_group}::A}
119 in part{constrs = constructors, rows = rows, A = []}
124 (*---------------------------------------------------------------------------
125 * Each pattern carries with it a tag (i,b) where
126 * i is the clause it came from and
127 * b=true indicates that clause was given by the user
128 * (or is an instantiation of a user supplied pattern)
130 *---------------------------------------------------------------------------*)
132 type pattern = term * (int * bool)
134 fun pattern_map f (tm,x) = (f tm, x);
136 fun pattern_subst theta = pattern_map (subst_free theta);
139 fun row_of_pat x = fst (snd x);
140 fun given x = snd (snd x);
142 (*---------------------------------------------------------------------------
143 * Produce an instance of a constructor, plus genvars for its arguments.
144 *---------------------------------------------------------------------------*)
145 fun fresh_constr ty_match colty gv c =
146 let val (_,Ty) = dest_Const c
147 val L = binder_types Ty
148 and ty = body_type Ty
149 val ty_theta = ty_match ty colty
150 val c' = S.inst ty_theta c
151 val gvars = map (S.inst ty_theta o gv) L
156 (*---------------------------------------------------------------------------
157 * Goes through a list of rows and picks out the ones beginning with a
158 * pattern with constructor = Name.
159 *---------------------------------------------------------------------------*)
160 fun mk_group Name rows =
161 U.itlist (fn (row as ((prfx, p::rst), rhs)) =>
162 fn (in_group,not_in_group) =>
163 let val (pc,args) = S.strip_comb p
164 in if ((#1 (Term.dest_Const pc) = Name) handle TERM _ => false)
165 then (((prfx,args@rst), rhs)::in_group, not_in_group)
166 else (in_group, row::not_in_group) end)
169 (*---------------------------------------------------------------------------
170 * Partition the rows. Not efficient: we should use hashing.
171 *---------------------------------------------------------------------------*)
172 fun partition _ _ (_,_,_,[]) = raise TFL_ERR "partition" "no rows"
173 | partition gv ty_match
174 (constructors, colty, res_ty, rows as (((prfx,_),_)::_)) =
175 let val fresh = fresh_constr ty_match colty gv
176 fun part {constrs = [], rows, A} = rev A
177 | part {constrs = c::crst, rows, A} =
178 let val (c',gvars) = fresh c
179 val (Name,Ty) = dest_Const c'
180 val (in_group, not_in_group) = mk_group Name rows
182 if (null in_group) (* Constructor not given *)
183 then [((prfx, #2(fresh c)), (S.ARB res_ty, (~1,false)))]
188 A = {constructor = c',
190 group = in_group'}::A}
192 in part{constrs=constructors, rows=rows, A=[]}
195 (*---------------------------------------------------------------------------
196 * Misc. routines used in mk_case
197 *---------------------------------------------------------------------------*)
200 let val L = length (binder_types (type_of c))
201 fun build (prfx,tag,plist) =
202 let val args = take (L,plist)
203 and plist' = drop(L,plist)
204 in (prfx,tag,list_comb(c,args)::plist') end
207 fun v_to_prfx (prfx, v::pats) = (v::prfx,pats)
208 | v_to_prfx _ = raise TFL_ERR "mk_case" "v_to_prfx";
210 fun v_to_pats (v::prfx,tag, pats) = (prfx, tag, v::pats)
211 | v_to_pats _ = raise TFL_ERR "mk_case" "v_to_pats";
214 (*----------------------------------------------------------------------------
215 * Translation of pattern terms into nested case expressions.
217 * This performs the translation and also builds the full set of patterns.
218 * Thus it supports the construction of induction theorems even when an
219 * incomplete set of patterns is given.
220 *---------------------------------------------------------------------------*)
222 fun mk_case ty_info ty_match usednames range_ty =
224 fun mk_case_fail s = raise TFL_ERR "mk_case" s
225 val fresh_var = gvvariant usednames
226 val divide = partition fresh_var ty_match
227 fun expand constructors ty ((_,[]), _) = mk_case_fail"expand_var_row"
228 | expand constructors ty (row as ((prfx, p::rst), rhs)) =
230 then let val fresh = fresh_constr ty_match ty fresh_var
232 let val capp = list_comb(c,gvs)
233 in ((prfx, capp::rst), pattern_subst[(p,capp)] rhs)
235 in map expnd (map fresh constructors) end
237 fun mk{rows=[],...} = mk_case_fail"no rows"
238 | mk{path=[], rows = ((prfx, []), (tm,tag))::_} = (* Done *)
239 ([(prfx,tag,[])], tm)
240 | mk{path=[], rows = _::_} = mk_case_fail"blunder"
241 | mk{path as u::rstp, rows as ((prfx, []), rhs)::rst} =
243 rows = ((prfx, [fresh_var(type_of u)]), rhs)::rst}
244 | mk{path = u::rstp, rows as ((_, p::_), _)::_} =
245 let val (pat_rectangle,rights) = ListPair.unzip rows
246 val col0 = map(hd o #2) pat_rectangle
248 if (forall is_Free col0)
249 then let val rights' = map (fn(v,e) => pattern_subst[(v,u)] e)
250 (ListPair.zip (col0, rights))
251 val pat_rectangle' = map v_to_prfx pat_rectangle
252 val (pref_patl,tm) = mk{path = rstp,
253 rows = ListPair.zip (pat_rectangle',
255 in (map v_to_pats pref_patl, tm)
258 let val pty as Type (ty_name,_) = type_of p
260 case (ty_info ty_name)
261 of None => mk_case_fail("Not a known datatype: "^ty_name)
262 | Some{case_const,constructors} =>
264 val case_const_name = #1(dest_Const case_const)
265 val nrows = List.concat (map (expand constructors pty) rows)
266 val subproblems = divide(constructors, pty, range_ty, nrows)
267 val groups = map #group subproblems
268 and new_formals = map #new_formals subproblems
269 and constructors' = map #constructor subproblems
270 val news = map (fn (nf,rows) => {path = nf@rstp, rows=rows})
271 (ListPair.zip (new_formals, groups))
272 val rec_calls = map mk news
273 val (pat_rect,dtrees) = ListPair.unzip rec_calls
274 val case_functions = map S.list_mk_abs
275 (ListPair.zip (new_formals, dtrees))
276 val types = map type_of (case_functions@[u]) @ [range_ty]
277 val case_const' = Const(case_const_name, list_mk_type types)
278 val tree = list_comb(case_const', case_functions@[u])
279 val pat_rect1 = List.concat
280 (ListPair.map mk_pat (constructors', pat_rect))
288 (* Repeated variable occurrences in a pattern are not allowed. *)
290 case (S.dest_term tm)
291 of S.VAR{Name,Ty} => [Free(Name,Ty)]
293 | S.COMB{Rator, Rand} => FV_multiset Rator @ FV_multiset Rand
294 | S.LAMB _ => raise TFL_ERR "FV_multiset" "lambda";
296 fun no_repeat_vars thy pat =
297 let fun check [] = true
299 if mem_term (v,rst) then
300 raise TFL_ERR "no_repeat_vars"
301 (quote (#1 (dest_Free v)) ^
302 " occurs repeatedly in the pattern " ^
303 quote (string_of_cterm (Thry.typecheck thy pat)))
305 in check (FV_multiset pat)
308 fun dest_atom (Free p) = p
309 | dest_atom (Const p) = p
310 | dest_atom _ = raise TFL_ERR "dest_atom" "function name not an identifier";
312 fun same_name (p,q) = #1(dest_atom p) = #1(dest_atom q);
314 local fun mk_functional_err s = raise TFL_ERR "mk_functional" s
316 mk_functional_err "recdef does not allow currying"
319 (*multiple function names?*)
320 if length (gen_distinct same_name fs) < length fs
321 then mk_functional_err
322 "The function being declared appears with multiple types"
323 else mk_functional_err
324 (Int.toString (length fs) ^
325 " distinct function names being declared")
327 fun mk_functional thy clauses =
328 let val (L,R) = ListPair.unzip (map HOLogic.dest_eq clauses
329 handle TERM _ => raise TFL_ERR "mk_functional"
330 "recursion equations must use the = relation")
331 val (funcs,pats) = ListPair.unzip (map (fn (t$u) =>(t,u)) L)
332 val atom = single (gen_distinct (op aconv) funcs)
333 val (fname,ftype) = dest_atom atom
334 val dummy = map (no_repeat_vars thy) pats
335 val rows = ListPair.zip (map (fn x => ([]:term list,[x])) pats,
336 map (fn (t,i) => (t,(i,true))) (enumerate R))
337 val names = foldr add_term_names (R,[])
338 val atype = type_of(hd pats)
339 and aname = variant names "a"
340 val a = Free(aname,atype)
341 val ty_info = Thry.match_info thy
342 val ty_match = Thry.match_type thy
343 val range_ty = type_of (hd R)
344 val (patts, case_tm) = mk_case ty_info ty_match (aname::names) range_ty
345 {path=[a], rows=rows}
346 val patts1 = map (fn (_,tag,[pat]) => (pat,tag)) patts
347 handle Match => mk_functional_err "error in pattern-match translation"
348 val patts2 = Library.sort (Library.int_ord o Library.pairself row_of_pat) patts1
349 val finals = map row_of_pat patts2
350 val originals = map (row_of_pat o #2) rows
351 val dummy = case (originals\\finals)
353 | L => mk_functional_err
354 ("The following clauses are redundant (covered by preceding clauses): " ^
355 commas (map (fn i => Int.toString (i + 1)) L))
356 in {functional = Abs(Sign.base_name fname, ftype,
358 absfree(aname,atype, case_tm))),
363 (*----------------------------------------------------------------------------
365 * PRINCIPLES OF DEFINITION
367 *---------------------------------------------------------------------------*)
370 (*For Isabelle, the lhs of a definition must be a constant.*)
371 fun mk_const_def sign (Name, Ty, rhs) =
372 Sign.infer_types sign (K None) (K None) [] false
373 ([Const("==",dummyT) $ Const(Name,Ty) $ rhs], propT)
376 (*Make all TVars available for instantiation by adding a ? to the front*)
377 fun poly_tvars (Type(a,Ts)) = Type(a, map (poly_tvars) Ts)
378 | poly_tvars (TFree (a,sort)) = TVar (("?" ^ a, 0), sort)
379 | poly_tvars (TVar ((a,i),sort)) = TVar (("?" ^ a, i+1), sort);
381 local val f_eq_wfrec_R_M =
382 #ant(S.dest_imp(#2(S.strip_forall (concl Thms.WFREC_COROLLARY))))
383 val {lhs=f, rhs} = S.dest_eq f_eq_wfrec_R_M
384 val (fname,_) = dest_Free f
385 val (wfrec,_) = S.strip_comb rhs
387 fun wfrec_definition0 thy fid R (functional as Abs(Name, Ty, _)) =
388 let val def_name = if Name<>fid then
389 raise TFL_ERR "wfrec_definition0"
390 ("Expected a definition of " ^
391 quote fid ^ " but found one of " ^
394 val wfrec_R_M = map_term_types poly_tvars
395 (wfrec $ map_term_types poly_tvars R)
397 val def_term = mk_const_def (Theory.sign_of thy) (Name, Ty, wfrec_R_M)
398 val (thy', [def]) = PureThy.add_defs_i false [Thm.no_attributes (def_name, def_term)] thy
404 (*---------------------------------------------------------------------------
405 * This structure keeps track of congruence rules that aren't derived
406 * from a datatype definition.
407 *---------------------------------------------------------------------------*)
408 fun extraction_thms thy =
409 let val {case_rewrites,case_congs} = Thry.extract_info thy
410 in (case_rewrites, case_congs)
414 (*---------------------------------------------------------------------------
415 * Pair patterns with termination conditions. The full list of patterns for
416 * a definition is merged with the TCs arising from the user-given clauses.
417 * There can be fewer clauses than the full list, if the user omitted some
418 * cases. This routine is used to prepare input for mk_induction.
419 *---------------------------------------------------------------------------*)
420 fun merge full_pats TCs =
421 let fun insert (p,TCs) =
422 let fun insrt ((x as (h,[]))::rst) =
423 if (p aconv h) then (p,TCs)::rst else x::insrt rst
424 | insrt (x::rst) = x::insrt rst
425 | insrt[] = raise TFL_ERR "merge.insert" "pattern not found"
427 fun pass ([],ptcl_final) = ptcl_final
428 | pass (ptcs::tcl, ptcl) = pass(tcl, insert ptcs ptcl)
430 pass (TCs, map (fn p => (p,[])) full_pats)
434 fun givens pats = map pat_of (filter given pats);
436 fun post_definition meta_tflCongs (theory, (def, pats)) =
437 let val tych = Thry.typecheck theory
438 val f = #lhs(S.dest_eq(concl def))
439 val corollary = R.MATCH_MP Thms.WFREC_COROLLARY def
440 val pats' = filter given pats
441 val given_pats = map pat_of pats'
442 val rows = map row_of_pat pats'
443 val WFR = #ant(S.dest_imp(concl corollary))
444 val R = #Rand(S.dest_comb WFR)
445 val corollary' = R.UNDISCH corollary (* put WF R on assums *)
446 val corollaries = map (fn pat => R.SPEC (tych pat) corollary')
448 val (case_rewrites,context_congs) = extraction_thms theory
449 val corollaries' = map(rewrite_rule case_rewrites) corollaries
450 val extract = R.CONTEXT_REWRITE_RULE
451 (f, [R], cut_apply, meta_tflCongs@context_congs)
452 val (rules, TCs) = ListPair.unzip (map extract corollaries')
453 val rules0 = map (rewrite_rule [Thms.CUT_DEF]) rules
454 val mk_cond_rule = R.FILTER_DISCH_ALL(not o curry (op aconv) WFR)
455 val rules1 = R.LIST_CONJ(map mk_cond_rule rules0)
460 full_pats_TCs = merge (map pat_of pats) (ListPair.zip (given_pats, TCs)),
465 (*---------------------------------------------------------------------------
466 * Perform the extraction without making the definition. Definition and
467 * extraction commute for the non-nested case. (Deferred recdefs)
469 * The purpose of wfrec_eqns is merely to instantiate the recursion theorem
470 * and extract termination conditions: no definition is made.
471 *---------------------------------------------------------------------------*)
473 fun wfrec_eqns thy fid tflCongs eqns =
474 let val {lhs,rhs} = S.dest_eq (hd eqns)
475 val (f,args) = S.strip_comb lhs
476 val (fname,fty) = dest_atom f
477 val (SV,a) = front_last args (* SV = schematic variables *)
478 val g = list_comb(f,SV)
479 val h = Free(fname,type_of g)
480 val eqns1 = map (subst_free[(g,h)]) eqns
481 val {functional as Abs(Name, Ty, _), pats} = mk_functional thy eqns1
482 val given_pats = givens pats
483 (* val f = Free(Name,Ty) *)
484 val Type("fun", [f_dty, f_rty]) = Ty
485 val dummy = if Name<>fid then
486 raise TFL_ERR "wfrec_eqns"
487 ("Expected a definition of " ^
488 quote fid ^ " but found one of " ^
491 val (case_rewrites,context_congs) = extraction_thms thy
492 val tych = Thry.typecheck thy
493 val WFREC_THM0 = R.ISPEC (tych functional) Thms.WFREC_COROLLARY
494 val Const("All",_) $ Abs(Rname,Rtype,_) = concl WFREC_THM0
495 val R = Free (variant (foldr add_term_names (eqns,[])) Rname,
497 val WFREC_THM = R.ISPECL [tych R, tych g] WFREC_THM0
498 val ([proto_def, WFR],_) = S.strip_imp(concl WFREC_THM)
501 writeln ("ORIGINAL PROTO_DEF: " ^
502 Sign.string_of_term (Theory.sign_of thy) proto_def)
505 val corollary' = R.UNDISCH(R.UNDISCH WFREC_THM)
506 val corollaries = map (fn pat => R.SPEC (tych pat) corollary') given_pats
507 val corollaries' = map (rewrite_rule case_rewrites) corollaries
508 fun extract X = R.CONTEXT_REWRITE_RULE
509 (f, R1::SV, cut_apply, tflCongs@context_congs) X
510 in {proto_def = proto_def,
514 extracta = map extract corollaries'}
518 (*---------------------------------------------------------------------------
519 * Define the constant after extracting the termination conditions. The
520 * wellfounded relation used in the definition is computed by using the
521 * choice operator on the extracted conditions (plus the condition that
522 * such a relation must be wellfounded).
523 *---------------------------------------------------------------------------*)
525 fun lazyR_def thy fid tflCongs eqns =
526 let val {proto_def,WFR,pats,extracta,SV} =
527 wfrec_eqns thy fid tflCongs eqns
529 val f = #lhs(S.dest_eq proto_def)
530 val (extractants,TCl) = ListPair.unzip extracta
531 val dummy = if !trace
532 then (writeln "Extractants = ";
536 val TCs = foldr (gen_union (op aconv)) (TCl, [])
537 val full_rqt = WFR::TCs
538 val R' = S.mk_select{Bvar=R1, Body=S.list_mk_conj full_rqt}
539 val R'abs = S.rand R'
540 val proto_def' = subst_free[(R1,R')] proto_def
541 val dummy = if !trace then writeln ("proto_def' = " ^
543 (Theory.sign_of thy) proto_def')
545 val {lhs,rhs} = S.dest_eq proto_def'
546 val (c,args) = S.strip_comb lhs
547 val (Name,Ty) = dest_atom c
548 val defn = mk_const_def (Theory.sign_of thy)
549 (Name, Ty, S.list_mk_abs (args,rhs))
550 val (theory, [def0]) =
552 |> PureThy.add_defs_i false
553 [Thm.no_attributes (fid ^ "_def", defn)]
554 val def = freezeT def0;
555 val dummy = if !trace then writeln ("DEF = " ^ string_of_thm def)
557 (* val fconst = #lhs(S.dest_eq(concl def)) *)
558 val tych = Thry.typecheck theory
559 val full_rqt_prop = map (Dcterm.mk_prop o tych) full_rqt
560 (*lcp: a lot of object-logic inference to remove*)
561 val baz = R.DISCH_ALL
562 (U.itlist R.DISCH full_rqt_prop
563 (R.LIST_CONJ extractants))
564 val dum = if !trace then writeln ("baz = " ^ string_of_thm baz)
566 val f_free = Free (fid, fastype_of f) (*'cos f is a Const*)
567 val SV' = map tych SV;
568 val SVrefls = map reflexive SV'
569 val def0 = (U.rev_itlist (fn x => fn th => R.rbeta(combination th x))
572 val def' = R.MP (R.SPEC (tych R') (R.GEN (tych R1) baz)) def0
573 val body_th = R.LIST_CONJ (map R.ASSUME full_rqt_prop)
574 val SELECT_AX = (*in this way we hope to avoid a STATIC dependence upon
575 theory Hilbert_Choice*)
576 thm "Hilbert_Choice.tfl_some"
577 handle ERROR => error
578 "defer_recdef requires theory Main or at least Hilbert_Choice as parent"
579 val bar = R.MP (R.ISPECL[tych R'abs, tych R1] SELECT_AX) body_th
580 in {theory = theory, R=R1, SV=SV,
581 rules = U.rev_itlist (U.C R.MP) (R.CONJUNCTS bar) def',
582 full_pats_TCs = merge (map pat_of pats) (ListPair.zip (givens pats, TCl)),
588 (*----------------------------------------------------------------------------
592 *---------------------------------------------------------------------------*)
595 (*------------------------ Miscellaneous function --------------------------
597 * [x_1,...,x_n] ?v_1...v_n. M[v_1,...,v_n]
598 * -----------------------------------------------------------
599 * ( M[x_1,...,x_n], [(x_i,?v_1...v_n. M[v_1,...,v_n]),
601 * (x_j,?v_n. M[x_1,...,x_(n-1),v_n])] )
603 * This function is totally ad hoc. Used in the production of the induction
604 * theorem. The nchotomy theorem can have clauses that look like
606 * ?v1..vn. z = C vn..v1
608 * in which the order of quantification is not the order of occurrence of the
609 * quantified variables as arguments to C. Since we have no control over this
610 * aspect of the nchotomy theorem, we make the correspondence explicit by
611 * pairing the incoming new variable with the term it gets beta-reduced into.
612 *---------------------------------------------------------------------------*)
614 fun alpha_ex_unroll (xlist, tm) =
615 let val (qvars,body) = S.strip_exists tm
616 val vlist = #2(S.strip_comb (S.rhs body))
617 val plist = ListPair.zip (vlist, xlist)
618 val args = map (fn qv => the (gen_assoc (op aconv) (plist, qv))) qvars
619 handle Library.OPTION => sys_error
620 "TFL fault [alpha_ex_unroll]: no correspondence"
622 | build (_$rex) (v::rst) =
623 let val ex1 = betapply(rex, v)
624 in ex1 :: build ex1 rst
626 val (nex::exl) = rev (tm::build tm args)
628 (nex, ListPair.zip (args, rev exl))
633 (*----------------------------------------------------------------------------
635 * PROVING COMPLETENESS OF PATTERNS
637 *---------------------------------------------------------------------------*)
639 fun mk_case ty_info usednames thy =
641 val divide = ipartition (gvvariant usednames)
642 val tych = Thry.typecheck thy
643 fun tych_binding(x,y) = (tych x, tych y)
644 fun fail s = raise TFL_ERR "mk_case" s
645 fun mk{rows=[],...} = fail"no rows"
646 | mk{path=[], rows = [([], (thm, bindings))]} =
647 R.IT_EXISTS (map tych_binding bindings) thm
648 | mk{path = u::rstp, rows as (p::_, _)::_} =
649 let val (pat_rectangle,rights) = ListPair.unzip rows
650 val col0 = map hd pat_rectangle
651 val pat_rectangle' = map tl pat_rectangle
653 if (forall is_Free col0) (* column 0 is all variables *)
654 then let val rights' = map (fn ((thm,theta),v) => (thm,theta@[(u,v)]))
655 (ListPair.zip (rights, col0))
656 in mk{path = rstp, rows = ListPair.zip (pat_rectangle', rights')}
658 else (* column 0 is all constructors *)
659 let val Type (ty_name,_) = type_of p
661 case (ty_info ty_name)
662 of None => fail("Not a known datatype: "^ty_name)
663 | Some{constructors,nchotomy} =>
664 let val thm' = R.ISPEC (tych u) nchotomy
665 val disjuncts = S.strip_disj (concl thm')
666 val subproblems = divide(constructors, rows)
667 val groups = map #group subproblems
668 and new_formals = map #new_formals subproblems
669 val existentials = ListPair.map alpha_ex_unroll
670 (new_formals, disjuncts)
671 val constraints = map #1 existentials
672 val vexl = map #2 existentials
673 fun expnd tm (pats,(th,b)) = (pats,(R.SUBS[R.ASSUME(tych tm)]th,b))
674 val news = map (fn (nf,rows,c) => {path = nf@rstp,
675 rows = map (expnd c) rows})
676 (U.zip3 new_formals groups constraints)
677 val recursive_thms = map mk news
678 val build_exists = foldr
680 R.CHOOSE (tych x, R.ASSUME (tych t)) th)
681 val thms' = ListPair.map build_exists (vexl, recursive_thms)
682 val same_concls = R.EVEN_ORS thms'
683 in R.DISJ_CASESL thm' same_concls
690 fun complete_cases thy =
691 let val tych = Thry.typecheck thy
692 val ty_info = Thry.induct_info thy
694 let val names = foldr add_term_names (pats,[])
695 val T = type_of (hd pats)
696 val aname = Term.variant names "a"
697 val vname = Term.variant (aname::names) "v"
698 val a = Free (aname, T)
699 val v = Free (vname, T)
700 val a_eq_v = HOLogic.mk_eq(a,v)
701 val ex_th0 = R.EXISTS (tych (S.mk_exists{Bvar=v,Body=a_eq_v}), tych a)
703 val th0 = R.ASSUME (tych a_eq_v)
704 val rows = map (fn x => ([x], (th0,[]))) pats
708 (R.CHOOSE(tych v, ex_th0)
709 (mk_case ty_info (vname::aname::names)
710 thy {path=[v], rows=rows})))
714 (*---------------------------------------------------------------------------
715 * Constructing induction hypotheses: one for each recursive call.
717 * Note. R will never occur as a variable in the ind_clause, because
718 * to do so, it would have to be from a nested definition, and we don't
719 * allow nested defns to have R variable.
721 * Note. When the context is empty, there can be no local variables.
722 *---------------------------------------------------------------------------*)
725 fun (tm1 ==> tm2) = S.mk_imp{ant = tm1, conseq = tm2}
727 fun build_ih f P (pat,TCs) =
728 let val globals = S.free_vars_lr pat
729 fun nested tm = is_some (S.find_term (curry (op aconv) f) tm)
731 let val (cntxt,R_y_pat) = S.strip_imp(#2(S.strip_forall tm))
732 val (R,y,_) = S.dest_relation R_y_pat
733 val P_y = if (nested tm) then R_y_pat ==> P$y else P$y
735 of [] => (P_y, (tm,[]))
737 val imp = S.list_mk_conj cntxt ==> P_y
738 val lvs = gen_rems (op aconv) (S.free_vars_lr imp, globals)
739 val locals = #2(U.pluck (curry (op aconv) P) lvs) handle U.ERR _ => lvs
740 in (S.list_mk_forall(locals,imp), (tm,locals)) end
743 of [] => (S.list_mk_forall(globals, P$pat), [])
744 | _ => let val (ihs, TCs_locals) = ListPair.unzip(map dest_TC TCs)
745 val ind_clause = S.list_mk_conj ihs ==> P$pat
746 in (S.list_mk_forall(globals,ind_clause), TCs_locals)
753 fun (tm1 ==> tm2) = S.mk_imp{ant = tm1, conseq = tm2}
755 fun build_ih f (P,SV) (pat,TCs) =
756 let val pat_vars = S.free_vars_lr pat
757 val globals = pat_vars@SV
758 fun nested tm = is_some (S.find_term (curry (op aconv) f) tm)
760 let val (cntxt,R_y_pat) = S.strip_imp(#2(S.strip_forall tm))
761 val (R,y,_) = S.dest_relation R_y_pat
762 val P_y = if (nested tm) then R_y_pat ==> P$y else P$y
764 of [] => (P_y, (tm,[]))
766 val imp = S.list_mk_conj cntxt ==> P_y
767 val lvs = gen_rems (op aconv) (S.free_vars_lr imp, globals)
768 val locals = #2(U.pluck (curry (op aconv) P) lvs) handle U.ERR _ => lvs
769 in (S.list_mk_forall(locals,imp), (tm,locals)) end
772 of [] => (S.list_mk_forall(pat_vars, P$pat), [])
773 | _ => let val (ihs, TCs_locals) = ListPair.unzip(map dest_TC TCs)
774 val ind_clause = S.list_mk_conj ihs ==> P$pat
775 in (S.list_mk_forall(pat_vars,ind_clause), TCs_locals)
780 (*---------------------------------------------------------------------------
781 * This function makes good on the promise made in "build_ih".
783 * Input is tm = "(!y. R y pat ==> P y) ==> P pat",
784 * TCs = TC_1[pat] ... TC_n[pat]
785 * thm = ih1 /\ ... /\ ih_n |- ih[pat]
786 *---------------------------------------------------------------------------*)
787 fun prove_case f thy (tm,TCs_locals,thm) =
788 let val tych = Thry.typecheck thy
789 val antc = tych(#ant(S.dest_imp tm))
790 val thm' = R.SPEC_ALL thm
791 fun nested tm = is_some (S.find_term (curry (op aconv) f) tm)
792 fun get_cntxt TC = tych(#ant(S.dest_imp(#2(S.strip_forall(concl TC)))))
793 fun mk_ih ((TC,locals),th2,nested) =
794 R.GENL (map tych locals)
795 (if nested then R.DISCH (get_cntxt TC) th2 handle U.ERR _ => th2
796 else if S.is_imp (concl TC) then R.IMP_TRANS TC th2
800 (if S.is_imp(concl thm') (* recursive calls in this clause *)
801 then let val th1 = R.ASSUME antc
802 val TCs = map #1 TCs_locals
803 val ylist = map (#2 o S.dest_relation o #2 o S.strip_imp o
804 #2 o S.strip_forall) TCs
805 val TClist = map (fn(TC,lvs) => (R.SPEC_ALL(R.ASSUME(tych TC)),lvs))
807 val th2list = map (U.C R.SPEC th1 o tych) ylist
808 val nlist = map nested TCs
809 val triples = U.zip3 TClist th2list nlist
810 val Pylist = map mk_ih triples
811 in R.MP thm' (R.LIST_CONJ Pylist) end
816 (*---------------------------------------------------------------------------
818 * x = (v1,...,vn) |- M[x]
819 * ---------------------------------------------
820 * ?v1 ... vn. x = (v1,...,vn) |- M[x]
822 *---------------------------------------------------------------------------*)
823 fun LEFT_ABS_VSTRUCT tych thm =
824 let fun CHOOSER v (tm,thm) =
825 let val ex_tm = S.mk_exists{Bvar=v,Body=tm}
826 in (ex_tm, R.CHOOSE(tych v, R.ASSUME (tych ex_tm)) thm)
828 val [veq] = filter (can S.dest_eq) (#1 (R.dest_thm thm))
829 val {lhs,rhs} = S.dest_eq veq
830 val L = S.free_vars_lr rhs
831 in #2 (U.itlist CHOOSER L (veq,thm)) end;
834 (*----------------------------------------------------------------------------
835 * Input : f, R, and [(pat1,TCs1),..., (patn,TCsn)]
837 * Instantiates WF_INDUCTION_THM, getting Sinduct and then tries to prove
838 * recursion induction (Rinduct) by proving the antecedent of Sinduct from
839 * the antecedent of Rinduct.
840 *---------------------------------------------------------------------------*)
841 fun mk_induction thy {fconst, R, SV, pat_TCs_list} =
842 let val tych = Thry.typecheck thy
843 val Sinduction = R.UNDISCH (R.ISPEC (tych R) Thms.WF_INDUCTION_THM)
844 val (pats,TCsl) = ListPair.unzip pat_TCs_list
845 val case_thm = complete_cases thy pats
846 val domain = (type_of o hd) pats
847 val Pname = Term.variant (foldr (foldr add_term_names)
848 (pats::TCsl, [])) "P"
849 val P = Free(Pname, domain --> HOLogic.boolT)
850 val Sinduct = R.SPEC (tych P) Sinduction
851 val Sinduct_assumf = S.rand ((#ant o S.dest_imp o concl) Sinduct)
852 val Rassums_TCl' = map (build_ih fconst (P,SV)) pat_TCs_list
853 val (Rassums,TCl') = ListPair.unzip Rassums_TCl'
854 val Rinduct_assum = R.ASSUME (tych (S.list_mk_conj Rassums))
855 val cases = map (fn pat => betapply (Sinduct_assumf, pat)) pats
856 val tasks = U.zip3 cases TCl' (R.CONJUNCTS Rinduct_assum)
857 val proved_cases = map (prove_case fconst thy) tasks
858 val v = Free (variant (foldr add_term_names (map concl proved_cases, []))
862 val substs = map (R.SYM o R.ASSUME o tych o (curry HOLogic.mk_eq v)) pats
863 val proved_cases1 = ListPair.map (fn (th,th') => R.SUBS[th]th')
864 (substs, proved_cases)
865 val abs_cases = map (LEFT_ABS_VSTRUCT tych) proved_cases1
866 val dant = R.GEN vtyped (R.DISJ_CASESL (R.ISPEC vtyped case_thm) abs_cases)
867 val dc = R.MP Sinduct dant
868 val Parg_ty = type_of(#Bvar(S.dest_forall(concl dc)))
869 val vars = map (gvvariant[Pname]) (S.strip_prod_type Parg_ty)
870 val dc' = U.itlist (R.GEN o tych) vars
871 (R.SPEC (tych(S.mk_vstruct Parg_ty vars)) dc)
873 R.GEN (tych P) (R.DISCH (tych(concl Rinduct_assum)) dc')
875 handle U.ERR _ => raise TFL_ERR "mk_induction" "failed derivation";
880 (*---------------------------------------------------------------------------
884 *---------------------------------------------------------------------------*)
887 fun simplify_induction thy hth ind =
888 let val tych = Thry.typecheck thy
889 val (asl,_) = R.dest_thm ind
890 val (_,tc_eq_tc') = R.dest_thm hth
891 val tc = S.lhs tc_eq_tc'
894 if (can (Thry.match_term thy asm) tc)
897 (R.MATCH_MP Thms.simp_thm (R.DISCH (tych asm) ind))
904 (*---------------------------------------------------------------------------
905 * The termination condition is an antecedent to the rule, and an
906 * assumption to the theorem.
907 *---------------------------------------------------------------------------*)
908 fun elim_tc tcthm (rule,induction) =
909 (R.MP rule tcthm, R.PROVE_HYP tcthm induction)
912 fun postprocess strict {wf_tac, terminator, simplifier} theory {rules,induction,TCs} =
913 let val tych = Thry.typecheck theory
914 val prove = R.prove strict;
916 (*---------------------------------------------------------------------
917 * Attempt to eliminate WF condition. It's the only assumption of rules
918 *---------------------------------------------------------------------*)
919 val (rules1,induction1) =
920 let val thm = prove(tych(HOLogic.mk_Trueprop
921 (hd(#1(R.dest_thm rules)))),
923 in (R.PROVE_HYP thm rules, R.PROVE_HYP thm induction)
924 end handle U.ERR _ => (rules,induction);
926 (*----------------------------------------------------------------------
927 * The termination condition (tc) is simplified to |- tc = tc' (there
928 * might not be a change!) and then 3 attempts are made:
930 * 1. if |- tc = T, then eliminate it with eqT; otherwise,
931 * 2. apply the terminator to tc'. If |- tc' = T then eliminate; else
932 * 3. replace tc by tc' in both the rules and the induction theorem.
933 *---------------------------------------------------------------------*)
936 if !trace then writeln (cat_lines (s :: map string_of_thm L))
939 fun print_cterms s L =
940 if !trace then writeln (cat_lines (s :: map string_of_cterm L))
943 fun simplify_tc tc (r,ind) =
944 let val tc1 = tych tc
945 val _ = print_cterms "TC before simplification: " [tc1]
946 val tc_eq = simplifier tc1
947 val _ = print_thms "result: " [tc_eq]
949 elim_tc (R.MATCH_MP Thms.eqT tc_eq) (r,ind)
951 (elim_tc (R.MATCH_MP(R.MATCH_MP Thms.rev_eq_mp tc_eq)
952 (prove(tych(HOLogic.mk_Trueprop(S.rhs(concl tc_eq))),
956 (R.UNDISCH(R.MATCH_MP (R.MATCH_MP Thms.simp_thm r) tc_eq),
957 simplify_induction theory tc_eq ind))
960 (*----------------------------------------------------------------------
961 * Nested termination conditions are harder to get at, since they are
962 * left embedded in the body of the function (and in induction
963 * theorem hypotheses). Our "solution" is to simplify them, and try to
964 * prove termination, but leave the application of the resulting theorem
965 * to a higher level. So things go much as in "simplify_tc": the
966 * termination condition (tc) is simplified to |- tc = tc' (there might
967 * not be a change) and then 2 attempts are made:
969 * 1. if |- tc = T, then return |- tc; otherwise,
970 * 2. apply the terminator to tc'. If |- tc' = T then return |- tc; else
971 * 3. return |- tc = tc'
972 *---------------------------------------------------------------------*)
973 fun simplify_nested_tc tc =
974 let val tc_eq = simplifier (tych (#2 (S.strip_forall tc)))
977 (R.MATCH_MP Thms.eqT tc_eq
979 (R.MATCH_MP(R.MATCH_MP Thms.rev_eq_mp tc_eq)
980 (prove(tych(HOLogic.mk_Trueprop (S.rhs(concl tc_eq))),
982 handle U.ERR _ => tc_eq))
985 (*-------------------------------------------------------------------
986 * Attempt to simplify the termination conditions in each rule and
987 * in the induction theorem.
988 *-------------------------------------------------------------------*)
989 fun strip_imp tm = if S.is_neg tm then ([],tm) else S.strip_imp tm
990 fun loop ([],extras,R,ind) = (rev R, ind, extras)
991 | loop ((r,ftcs)::rst, nthms, R, ind) =
992 let val tcs = #1(strip_imp (concl r))
993 val extra_tcs = gen_rems (op aconv) (ftcs, tcs)
994 val extra_tc_thms = map simplify_nested_tc extra_tcs
995 val (r1,ind1) = U.rev_itlist simplify_tc tcs (r,ind)
996 val r2 = R.FILTER_DISCH_ALL(not o S.is_WFR) r1
997 in loop(rst, nthms@extra_tc_thms, r2::R, ind1)
999 val rules_tcs = ListPair.zip (R.CONJUNCTS rules1, TCs)
1000 val (rules2,ind2,extras) = loop(rules_tcs,[],[],induction1)
1002 {induction = ind2, rules = R.LIST_CONJ rules2, nested_tcs = extras}