1 (* Title: HOL/Tools/TFL/tfl.ML
2 Author: Konrad Slind, Cambridge University Computer Laboratory
4 First part of main module.
10 val trace_thms: string -> thm list -> unit
11 val trace_cterms: string -> cterm list -> unit
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) ->
19 full_pats_TCs: (term * term list) list}
20 val wfrec_eqns: theory -> xstring -> thm list -> term list ->
24 extracta: (thm * term list) list,
26 val lazyR_def: theory -> xstring -> thm list -> term list ->
31 full_pats_TCs: (term * term list) list,
32 patterns : pattern list}
33 val mk_induction: theory ->
34 {fconst: term, R: term, SV: term list, pat_TCs_list: (term * term list) list} -> thm
35 val postprocess: bool -> {wf_tac: tactic, terminator: tactic, simplifier: cterm -> thm}
36 -> theory -> {rules: thm, induction: thm, TCs: term list list}
37 -> {rules: thm, induction: thm, nested_tcs: thm list}
40 structure Prim: PRIM =
43 val trace = ref false;
46 structure S = USyntax;
50 fun TFL_ERR func mesg = U.ERR {module = "Tfl", func = func, mesg = mesg};
52 val concl = #2 o R.dest_thm;
53 val hyp = #1 o R.dest_thm;
55 val list_mk_type = U.end_itlist (curry (op -->));
57 fun enumerate xs = ListPair.zip(xs, 0 upto (length xs - 1));
59 fun front_last [] = raise TFL_ERR "front_last" "empty list"
60 | front_last [x] = ([],x)
62 let val (pref,x) = front_last t
68 (*---------------------------------------------------------------------------
69 * The next function is common to pattern-match translation and
70 * proof of completeness of cases for the induction theorem.
72 * The curried function "gvvariant" returns a function to generate distinct
73 * variables that are guaranteed not to be in names. The names of
74 * the variables go u, v, ..., z, aa, ..., az, ... The returned
75 * function contains embedded refs!
76 *---------------------------------------------------------------------------*)
78 let val slist = ref names
81 if !vname mem_string (!slist)
82 then (vname := Symbol.bump_string (!vname); new())
83 else (slist := !vname :: !slist; !vname)
85 fn ty => Free(new(), ty)
89 (*---------------------------------------------------------------------------
90 * Used in induction theorem production. This is the simple case of
91 * partitioning up pattern rows by the leading constructor.
92 *---------------------------------------------------------------------------*)
93 fun ipartition gv (constructors,rows) =
94 let fun pfail s = raise TFL_ERR "partition.part" s
95 fun part {constrs = [], rows = [], A} = rev A
96 | part {constrs = [], rows = _::_, A} = pfail"extra cases in defn"
97 | part {constrs = _::_, rows = [], A} = pfail"cases missing in defn"
98 | part {constrs = c::crst, rows, A} =
99 let val (c, T) = dest_Const c
100 val L = binder_types T
101 val (in_group, not_in_group) =
102 fold_rev (fn (row as (p::rst, rhs)) =>
103 fn (in_group,not_in_group) =>
104 let val (pc,args) = S.strip_comb p
105 in if (#1(dest_Const pc) = c)
106 then ((args@rst, rhs)::in_group, not_in_group)
107 else (in_group, row::not_in_group)
109 val col_types = U.take type_of (length L, #1(hd in_group))
111 part{constrs = crst, rows = not_in_group,
112 A = {constructor = c,
113 new_formals = map gv col_types,
114 group = in_group}::A}
116 in part{constrs = constructors, rows = rows, A = []}
121 (*---------------------------------------------------------------------------
122 * Each pattern carries with it a tag (i,b) where
123 * i is the clause it came from and
124 * b=true indicates that clause was given by the user
125 * (or is an instantiation of a user supplied pattern)
127 *---------------------------------------------------------------------------*)
129 type pattern = term * (int * bool)
131 fun pattern_map f (tm,x) = (f tm, x);
133 fun pattern_subst theta = pattern_map (subst_free theta);
136 fun row_of_pat x = fst (snd x);
137 fun given x = snd (snd x);
139 (*---------------------------------------------------------------------------
140 * Produce an instance of a constructor, plus genvars for its arguments.
141 *---------------------------------------------------------------------------*)
142 fun fresh_constr ty_match colty gv c =
143 let val (_,Ty) = dest_Const c
144 val L = binder_types Ty
145 and ty = body_type Ty
146 val ty_theta = ty_match ty colty
147 val c' = S.inst ty_theta c
148 val gvars = map (S.inst ty_theta o gv) L
153 (*---------------------------------------------------------------------------
154 * Goes through a list of rows and picks out the ones beginning with a
155 * pattern with constructor = name.
156 *---------------------------------------------------------------------------*)
157 fun mk_group name rows =
158 fold_rev (fn (row as ((prfx, p::rst), rhs)) =>
159 fn (in_group,not_in_group) =>
160 let val (pc,args) = S.strip_comb p
161 in if ((#1 (Term.dest_Const pc) = name) handle TERM _ => false)
162 then (((prfx,args@rst), rhs)::in_group, not_in_group)
163 else (in_group, row::not_in_group) end)
166 (*---------------------------------------------------------------------------
167 * Partition the rows. Not efficient: we should use hashing.
168 *---------------------------------------------------------------------------*)
169 fun partition _ _ (_,_,_,[]) = raise TFL_ERR "partition" "no rows"
170 | partition gv ty_match
171 (constructors, colty, res_ty, rows as (((prfx,_),_)::_)) =
172 let val fresh = fresh_constr ty_match colty gv
173 fun part {constrs = [], rows, A} = rev A
174 | part {constrs = c::crst, rows, A} =
175 let val (c',gvars) = fresh c
176 val (in_group, not_in_group) = mk_group (#1 (dest_Const c')) rows
178 if (null in_group) (* Constructor not given *)
179 then [((prfx, #2(fresh c)), (S.ARB res_ty, (~1,false)))]
184 A = {constructor = c',
186 group = in_group'}::A}
188 in part{constrs=constructors, rows=rows, A=[]}
191 (*---------------------------------------------------------------------------
192 * Misc. routines used in mk_case
193 *---------------------------------------------------------------------------*)
196 let val L = length (binder_types (type_of c))
197 fun build (prfx,tag,plist) =
198 let val args = Library.take (L,plist)
199 and plist' = Library.drop(L,plist)
200 in (prfx,tag,list_comb(c,args)::plist') end
203 fun v_to_prfx (prfx, v::pats) = (v::prfx,pats)
204 | v_to_prfx _ = raise TFL_ERR "mk_case" "v_to_prfx";
206 fun v_to_pats (v::prfx,tag, pats) = (prfx, tag, v::pats)
207 | v_to_pats _ = raise TFL_ERR "mk_case" "v_to_pats";
210 (*----------------------------------------------------------------------------
211 * Translation of pattern terms into nested case expressions.
213 * This performs the translation and also builds the full set of patterns.
214 * Thus it supports the construction of induction theorems even when an
215 * incomplete set of patterns is given.
216 *---------------------------------------------------------------------------*)
218 fun mk_case ty_info ty_match usednames range_ty =
220 fun mk_case_fail s = raise TFL_ERR "mk_case" s
221 val fresh_var = gvvariant usednames
222 val divide = partition fresh_var ty_match
223 fun expand constructors ty ((_,[]), _) = mk_case_fail"expand_var_row"
224 | expand constructors ty (row as ((prfx, p::rst), rhs)) =
226 then let val fresh = fresh_constr ty_match ty fresh_var
228 let val capp = list_comb(c,gvs)
229 in ((prfx, capp::rst), pattern_subst[(p,capp)] rhs)
231 in map expnd (map fresh constructors) end
233 fun mk{rows=[],...} = mk_case_fail"no rows"
234 | mk{path=[], rows = ((prfx, []), (tm,tag))::_} = (* Done *)
235 ([(prfx,tag,[])], tm)
236 | mk{path=[], rows = _::_} = mk_case_fail"blunder"
237 | mk{path as u::rstp, rows as ((prfx, []), rhs)::rst} =
239 rows = ((prfx, [fresh_var(type_of u)]), rhs)::rst}
240 | mk{path = u::rstp, rows as ((_, p::_), _)::_} =
241 let val (pat_rectangle,rights) = ListPair.unzip rows
242 val col0 = map(hd o #2) pat_rectangle
244 if (forall is_Free col0)
245 then let val rights' = map (fn(v,e) => pattern_subst[(v,u)] e)
246 (ListPair.zip (col0, rights))
247 val pat_rectangle' = map v_to_prfx pat_rectangle
248 val (pref_patl,tm) = mk{path = rstp,
249 rows = ListPair.zip (pat_rectangle',
251 in (map v_to_pats pref_patl, tm)
254 let val pty as Type (ty_name,_) = type_of p
256 case (ty_info ty_name)
257 of NONE => mk_case_fail("Not a known datatype: "^ty_name)
258 | SOME{case_const,constructors} =>
260 val case_const_name = #1(dest_Const case_const)
261 val nrows = List.concat (map (expand constructors pty) rows)
262 val subproblems = divide(constructors, pty, range_ty, nrows)
263 val groups = map #group subproblems
264 and new_formals = map #new_formals subproblems
265 and constructors' = map #constructor subproblems
266 val news = map (fn (nf,rows) => {path = nf@rstp, rows=rows})
267 (ListPair.zip (new_formals, groups))
268 val rec_calls = map mk news
269 val (pat_rect,dtrees) = ListPair.unzip rec_calls
270 val case_functions = map S.list_mk_abs
271 (ListPair.zip (new_formals, dtrees))
272 val types = map type_of (case_functions@[u]) @ [range_ty]
273 val case_const' = Const(case_const_name, list_mk_type types)
274 val tree = list_comb(case_const', case_functions@[u])
275 val pat_rect1 = List.concat
276 (ListPair.map mk_pat (constructors', pat_rect))
284 (* Repeated variable occurrences in a pattern are not allowed. *)
286 case (S.dest_term tm)
287 of S.VAR{Name = c, Ty = T} => [Free(c, T)]
289 | S.COMB{Rator, Rand} => FV_multiset Rator @ FV_multiset Rand
290 | S.LAMB _ => raise TFL_ERR "FV_multiset" "lambda";
292 fun no_repeat_vars thy pat =
293 let fun check [] = true
295 if member (op aconv) rst v then
296 raise TFL_ERR "no_repeat_vars"
297 (quote (#1 (dest_Free v)) ^
298 " occurs repeatedly in the pattern " ^
299 quote (Display.string_of_cterm (Thry.typecheck thy pat)))
301 in check (FV_multiset pat)
304 fun dest_atom (Free p) = p
305 | dest_atom (Const p) = p
306 | dest_atom _ = raise TFL_ERR "dest_atom" "function name not an identifier";
308 fun same_name (p,q) = #1(dest_atom p) = #1(dest_atom q);
310 local fun mk_functional_err s = raise TFL_ERR "mk_functional" s
312 mk_functional_err "recdef does not allow currying"
315 (*multiple function names?*)
316 if length (distinct same_name fs) < length fs
317 then mk_functional_err
318 "The function being declared appears with multiple types"
319 else mk_functional_err
320 (Int.toString (length fs) ^
321 " distinct function names being declared")
323 fun mk_functional thy clauses =
324 let val (L,R) = ListPair.unzip (map HOLogic.dest_eq clauses
325 handle TERM _ => raise TFL_ERR "mk_functional"
326 "recursion equations must use the = relation")
327 val (funcs,pats) = ListPair.unzip (map (fn (t$u) =>(t,u)) L)
328 val atom = single (distinct (op aconv) funcs)
329 val (fname,ftype) = dest_atom atom
330 val dummy = map (no_repeat_vars thy) pats
331 val rows = ListPair.zip (map (fn x => ([]:term list,[x])) pats,
332 map (fn (t,i) => (t,(i,true))) (enumerate R))
333 val names = List.foldr OldTerm.add_term_names [] R
334 val atype = type_of(hd pats)
335 and aname = Name.variant names "a"
336 val a = Free(aname,atype)
337 val ty_info = Thry.match_info thy
338 val ty_match = Thry.match_type thy
339 val range_ty = type_of (hd R)
340 val (patts, case_tm) = mk_case ty_info ty_match (aname::names) range_ty
341 {path=[a], rows=rows}
342 val patts1 = map (fn (_,tag,[pat]) => (pat,tag)) patts
343 handle Match => mk_functional_err "error in pattern-match translation"
344 val patts2 = Library.sort (Library.int_ord o Library.pairself row_of_pat) patts1
345 val finals = map row_of_pat patts2
346 val originals = map (row_of_pat o #2) rows
347 val dummy = case (originals\\finals)
349 | L => mk_functional_err
350 ("The following clauses are redundant (covered by preceding clauses): " ^
351 commas (map (fn i => Int.toString (i + 1)) L))
352 in {functional = Abs(NameSpace.base_name fname, ftype,
354 absfree(aname,atype, case_tm))),
359 (*----------------------------------------------------------------------------
361 * PRINCIPLES OF DEFINITION
363 *---------------------------------------------------------------------------*)
366 (*For Isabelle, the lhs of a definition must be a constant.*)
367 fun mk_const_def sign (c, Ty, rhs) =
368 singleton (Syntax.check_terms (ProofContext.init sign))
369 (Sign.intern_term sign (Const("==",dummyT) $ Const(c,Ty) $ rhs));
371 (*Make all TVars available for instantiation by adding a ? to the front*)
372 fun poly_tvars (Type(a,Ts)) = Type(a, map (poly_tvars) Ts)
373 | poly_tvars (TFree (a,sort)) = TVar (("?" ^ a, 0), sort)
374 | poly_tvars (TVar ((a,i),sort)) = TVar (("?" ^ a, i+1), sort);
376 local val f_eq_wfrec_R_M =
377 #ant(S.dest_imp(#2(S.strip_forall (concl Thms.WFREC_COROLLARY))))
378 val {lhs=f, rhs} = S.dest_eq f_eq_wfrec_R_M
379 val (fname,_) = dest_Free f
380 val (wfrec,_) = S.strip_comb rhs
382 fun wfrec_definition0 thy fid R (functional as Abs(x, Ty, _)) =
383 let val def_name = if x<>fid then
384 raise TFL_ERR "wfrec_definition0"
385 ("Expected a definition of " ^
386 quote fid ^ " but found one of " ^
389 val wfrec_R_M = map_types poly_tvars
390 (wfrec $ map_types poly_tvars R)
392 val def_term = mk_const_def thy (x, Ty, wfrec_R_M)
393 val ([def], thy') = PureThy.add_defs false [Thm.no_attributes (Binding.name def_name, def_term)] thy
399 (*---------------------------------------------------------------------------
400 * This structure keeps track of congruence rules that aren't derived
401 * from a datatype definition.
402 *---------------------------------------------------------------------------*)
403 fun extraction_thms thy =
404 let val {case_rewrites,case_congs} = Thry.extract_info thy
405 in (case_rewrites, case_congs)
409 (*---------------------------------------------------------------------------
410 * Pair patterns with termination conditions. The full list of patterns for
411 * a definition is merged with the TCs arising from the user-given clauses.
412 * There can be fewer clauses than the full list, if the user omitted some
413 * cases. This routine is used to prepare input for mk_induction.
414 *---------------------------------------------------------------------------*)
415 fun merge full_pats TCs =
416 let fun insert (p,TCs) =
417 let fun insrt ((x as (h,[]))::rst) =
418 if (p aconv h) then (p,TCs)::rst else x::insrt rst
419 | insrt (x::rst) = x::insrt rst
420 | insrt[] = raise TFL_ERR "merge.insert" "pattern not found"
422 fun pass ([],ptcl_final) = ptcl_final
423 | pass (ptcs::tcl, ptcl) = pass(tcl, insert ptcs ptcl)
425 pass (TCs, map (fn p => (p,[])) full_pats)
429 fun givens pats = map pat_of (List.filter given pats);
431 fun post_definition meta_tflCongs (theory, (def, pats)) =
432 let val tych = Thry.typecheck theory
433 val f = #lhs(S.dest_eq(concl def))
434 val corollary = R.MATCH_MP Thms.WFREC_COROLLARY def
435 val pats' = List.filter given pats
436 val given_pats = map pat_of pats'
437 val rows = map row_of_pat pats'
438 val WFR = #ant(S.dest_imp(concl corollary))
439 val R = #Rand(S.dest_comb WFR)
440 val corollary' = R.UNDISCH corollary (* put WF R on assums *)
441 val corollaries = map (fn pat => R.SPEC (tych pat) corollary')
443 val (case_rewrites,context_congs) = extraction_thms theory
444 (*case_ss causes minimal simplification: bodies of case expressions are
445 not simplified. Otherwise large examples (Red-Black trees) are too
447 val case_ss = Simplifier.theory_context theory
448 (HOL_basic_ss addcongs
449 (map (#weak_case_cong o snd) o Symtab.dest o DatatypePackage.get_datatypes) theory addsimps case_rewrites)
450 val corollaries' = map (Simplifier.simplify case_ss) corollaries
451 val extract = R.CONTEXT_REWRITE_RULE
452 (f, [R], @{thm cut_apply}, meta_tflCongs@context_congs)
453 val (rules, TCs) = ListPair.unzip (map extract corollaries')
454 val rules0 = map (rewrite_rule [Thms.CUT_DEF]) rules
455 val mk_cond_rule = R.FILTER_DISCH_ALL(not o curry (op aconv) WFR)
456 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(x, Ty, _), pats} = mk_functional thy eqns1
482 val given_pats = givens pats
483 (* val f = Free(x,Ty) *)
484 val Type("fun", [f_dty, f_rty]) = Ty
485 val dummy = if x<>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 (Name.variant (List.foldr OldTerm.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 Syntax.string_of_term_global 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, @{thm 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 = ";
533 Display.prths extractants;
536 val TCs = List.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' = " ^
542 Syntax.string_of_term_global
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 thy (name, Ty, S.list_mk_abs (args,rhs))
549 val ([def0], theory) =
551 |> PureThy.add_defs false
552 [Thm.no_attributes (Binding.name (fid ^ "_def"), defn)]
553 val def = Thm.freezeT def0;
554 val dummy = if !trace then writeln ("DEF = " ^ Display.string_of_thm def)
556 (* val fconst = #lhs(S.dest_eq(concl def)) *)
557 val tych = Thry.typecheck theory
558 val full_rqt_prop = map (Dcterm.mk_prop o tych) full_rqt
559 (*lcp: a lot of object-logic inference to remove*)
560 val baz = R.DISCH_ALL
561 (fold_rev R.DISCH full_rqt_prop
562 (R.LIST_CONJ extractants))
563 val dum = if !trace then writeln ("baz = " ^ Display.string_of_thm baz)
565 val f_free = Free (fid, fastype_of f) (*'cos f is a Const*)
566 val SV' = map tych SV;
567 val SVrefls = map reflexive SV'
568 val def0 = (fold (fn x => fn th => R.rbeta(combination th x))
571 val def' = R.MP (R.SPEC (tych R') (R.GEN (tych R1) baz)) def0
572 val body_th = R.LIST_CONJ (map R.ASSUME full_rqt_prop)
573 val SELECT_AX = (*in this way we hope to avoid a STATIC dependence upon
574 theory Hilbert_Choice*)
575 thm "Hilbert_Choice.tfl_some"
576 handle ERROR msg => cat_error msg
577 "defer_recdef requires theory Main or at least Hilbert_Choice as parent"
578 val bar = R.MP (R.ISPECL[tych R'abs, tych R1] SELECT_AX) body_th
579 in {theory = theory, R=R1, SV=SV,
580 rules = fold (U.C R.MP) (R.CONJUNCTS bar) def',
581 full_pats_TCs = merge (map pat_of pats) (ListPair.zip (givens pats, TCl)),
587 (*----------------------------------------------------------------------------
591 *---------------------------------------------------------------------------*)
594 (*------------------------ Miscellaneous function --------------------------
596 * [x_1,...,x_n] ?v_1...v_n. M[v_1,...,v_n]
597 * -----------------------------------------------------------
598 * ( M[x_1,...,x_n], [(x_i,?v_1...v_n. M[v_1,...,v_n]),
600 * (x_j,?v_n. M[x_1,...,x_(n-1),v_n])] )
602 * This function is totally ad hoc. Used in the production of the induction
603 * theorem. The nchotomy theorem can have clauses that look like
605 * ?v1..vn. z = C vn..v1
607 * in which the order of quantification is not the order of occurrence of the
608 * quantified variables as arguments to C. Since we have no control over this
609 * aspect of the nchotomy theorem, we make the correspondence explicit by
610 * pairing the incoming new variable with the term it gets beta-reduced into.
611 *---------------------------------------------------------------------------*)
613 fun alpha_ex_unroll (xlist, tm) =
614 let val (qvars,body) = S.strip_exists tm
615 val vlist = #2(S.strip_comb (S.rhs body))
616 val plist = ListPair.zip (vlist, xlist)
617 val args = map (the o AList.lookup (op aconv) plist) qvars
618 handle Option => sys_error
619 "TFL fault [alpha_ex_unroll]: no correspondence"
621 | build (_$rex) (v::rst) =
622 let val ex1 = Term.betapply(rex, v)
623 in ex1 :: build ex1 rst
625 val (nex::exl) = rev (tm::build tm args)
627 (nex, ListPair.zip (args, rev exl))
632 (*----------------------------------------------------------------------------
634 * PROVING COMPLETENESS OF PATTERNS
636 *---------------------------------------------------------------------------*)
638 fun mk_case ty_info usednames thy =
640 val divide = ipartition (gvvariant usednames)
641 val tych = Thry.typecheck thy
642 fun tych_binding(x,y) = (tych x, tych y)
643 fun fail s = raise TFL_ERR "mk_case" s
644 fun mk{rows=[],...} = fail"no rows"
645 | mk{path=[], rows = [([], (thm, bindings))]} =
646 R.IT_EXISTS (map tych_binding bindings) thm
647 | mk{path = u::rstp, rows as (p::_, _)::_} =
648 let val (pat_rectangle,rights) = ListPair.unzip rows
649 val col0 = map hd pat_rectangle
650 val pat_rectangle' = map tl pat_rectangle
652 if (forall is_Free col0) (* column 0 is all variables *)
653 then let val rights' = map (fn ((thm,theta),v) => (thm,theta@[(u,v)]))
654 (ListPair.zip (rights, col0))
655 in mk{path = rstp, rows = ListPair.zip (pat_rectangle', rights')}
657 else (* column 0 is all constructors *)
658 let val Type (ty_name,_) = type_of p
660 case (ty_info ty_name)
661 of NONE => fail("Not a known datatype: "^ty_name)
662 | SOME{constructors,nchotomy} =>
663 let val thm' = R.ISPEC (tych u) nchotomy
664 val disjuncts = S.strip_disj (concl thm')
665 val subproblems = divide(constructors, rows)
666 val groups = map #group subproblems
667 and new_formals = map #new_formals subproblems
668 val existentials = ListPair.map alpha_ex_unroll
669 (new_formals, disjuncts)
670 val constraints = map #1 existentials
671 val vexl = map #2 existentials
672 fun expnd tm (pats,(th,b)) = (pats,(R.SUBS[R.ASSUME(tych tm)]th,b))
673 val news = map (fn (nf,rows,c) => {path = nf@rstp,
674 rows = map (expnd c) rows})
675 (U.zip3 new_formals groups constraints)
676 val recursive_thms = map mk news
677 val build_exists = Library.foldr
679 R.CHOOSE (tych x, R.ASSUME (tych t)) th)
680 val thms' = ListPair.map build_exists (vexl, recursive_thms)
681 val same_concls = R.EVEN_ORS thms'
682 in R.DISJ_CASESL thm' same_concls
689 fun complete_cases thy =
690 let val tych = Thry.typecheck thy
691 val ty_info = Thry.induct_info thy
693 let val names = List.foldr OldTerm.add_term_names [] pats
694 val T = type_of (hd pats)
695 val aname = Name.variant names "a"
696 val vname = Name.variant (aname::names) "v"
697 val a = Free (aname, T)
698 val v = Free (vname, T)
699 val a_eq_v = HOLogic.mk_eq(a,v)
700 val ex_th0 = R.EXISTS (tych (S.mk_exists{Bvar=v,Body=a_eq_v}), tych a)
702 val th0 = R.ASSUME (tych a_eq_v)
703 val rows = map (fn x => ([x], (th0,[]))) pats
707 (R.CHOOSE(tych v, ex_th0)
708 (mk_case ty_info (vname::aname::names)
709 thy {path=[v], rows=rows})))
713 (*---------------------------------------------------------------------------
714 * Constructing induction hypotheses: one for each recursive call.
716 * Note. R will never occur as a variable in the ind_clause, because
717 * to do so, it would have to be from a nested definition, and we don't
718 * allow nested defns to have R variable.
720 * Note. When the context is empty, there can be no local variables.
721 *---------------------------------------------------------------------------*)
724 fun (tm1 ==> tm2) = S.mk_imp{ant = tm1, conseq = tm2}
726 fun build_ih f P (pat,TCs) =
727 let val globals = S.free_vars_lr pat
728 fun nested tm = isSome (S.find_term (curry (op aconv) f) tm)
730 let val (cntxt,R_y_pat) = S.strip_imp(#2(S.strip_forall tm))
731 val (R,y,_) = S.dest_relation R_y_pat
732 val P_y = if (nested tm) then R_y_pat ==> P$y else P$y
734 of [] => (P_y, (tm,[]))
736 val imp = S.list_mk_conj cntxt ==> P_y
737 val lvs = gen_rems (op aconv) (S.free_vars_lr imp, globals)
738 val locals = #2(U.pluck (curry (op aconv) P) lvs) handle U.ERR _ => lvs
739 in (S.list_mk_forall(locals,imp), (tm,locals)) end
742 of [] => (S.list_mk_forall(globals, P$pat), [])
743 | _ => let val (ihs, TCs_locals) = ListPair.unzip(map dest_TC TCs)
744 val ind_clause = S.list_mk_conj ihs ==> P$pat
745 in (S.list_mk_forall(globals,ind_clause), TCs_locals)
752 fun (tm1 ==> tm2) = S.mk_imp{ant = tm1, conseq = tm2}
754 fun build_ih f (P,SV) (pat,TCs) =
755 let val pat_vars = S.free_vars_lr pat
756 val globals = pat_vars@SV
757 fun nested tm = isSome (S.find_term (curry (op aconv) f) tm)
759 let val (cntxt,R_y_pat) = S.strip_imp(#2(S.strip_forall tm))
760 val (R,y,_) = S.dest_relation R_y_pat
761 val P_y = if (nested tm) then R_y_pat ==> P$y else P$y
763 of [] => (P_y, (tm,[]))
765 val imp = S.list_mk_conj cntxt ==> P_y
766 val lvs = subtract (op aconv) globals (S.free_vars_lr imp)
767 val locals = #2(U.pluck (curry (op aconv) P) lvs) handle U.ERR _ => lvs
768 in (S.list_mk_forall(locals,imp), (tm,locals)) end
771 of [] => (S.list_mk_forall(pat_vars, P$pat), [])
772 | _ => let val (ihs, TCs_locals) = ListPair.unzip(map dest_TC TCs)
773 val ind_clause = S.list_mk_conj ihs ==> P$pat
774 in (S.list_mk_forall(pat_vars,ind_clause), TCs_locals)
779 (*---------------------------------------------------------------------------
780 * This function makes good on the promise made in "build_ih".
782 * Input is tm = "(!y. R y pat ==> P y) ==> P pat",
783 * TCs = TC_1[pat] ... TC_n[pat]
784 * thm = ih1 /\ ... /\ ih_n |- ih[pat]
785 *---------------------------------------------------------------------------*)
786 fun prove_case f thy (tm,TCs_locals,thm) =
787 let val tych = Thry.typecheck thy
788 val antc = tych(#ant(S.dest_imp tm))
789 val thm' = R.SPEC_ALL thm
790 fun nested tm = isSome (S.find_term (curry (op aconv) f) tm)
791 fun get_cntxt TC = tych(#ant(S.dest_imp(#2(S.strip_forall(concl TC)))))
792 fun mk_ih ((TC,locals),th2,nested) =
793 R.GENL (map tych locals)
794 (if nested then R.DISCH (get_cntxt TC) th2 handle U.ERR _ => th2
795 else if S.is_imp (concl TC) then R.IMP_TRANS TC th2
799 (if S.is_imp(concl thm') (* recursive calls in this clause *)
800 then let val th1 = R.ASSUME antc
801 val TCs = map #1 TCs_locals
802 val ylist = map (#2 o S.dest_relation o #2 o S.strip_imp o
803 #2 o S.strip_forall) TCs
804 val TClist = map (fn(TC,lvs) => (R.SPEC_ALL(R.ASSUME(tych TC)),lvs))
806 val th2list = map (U.C R.SPEC th1 o tych) ylist
807 val nlist = map nested TCs
808 val triples = U.zip3 TClist th2list nlist
809 val Pylist = map mk_ih triples
810 in R.MP thm' (R.LIST_CONJ Pylist) end
815 (*---------------------------------------------------------------------------
817 * x = (v1,...,vn) |- M[x]
818 * ---------------------------------------------
819 * ?v1 ... vn. x = (v1,...,vn) |- M[x]
821 *---------------------------------------------------------------------------*)
822 fun LEFT_ABS_VSTRUCT tych thm =
823 let fun CHOOSER v (tm,thm) =
824 let val ex_tm = S.mk_exists{Bvar=v,Body=tm}
825 in (ex_tm, R.CHOOSE(tych v, R.ASSUME (tych ex_tm)) thm)
827 val [veq] = List.filter (can S.dest_eq) (#1 (R.dest_thm thm))
828 val {lhs,rhs} = S.dest_eq veq
829 val L = S.free_vars_lr rhs
830 in #2 (fold_rev CHOOSER L (veq,thm)) end;
833 (*----------------------------------------------------------------------------
834 * Input : f, R, and [(pat1,TCs1),..., (patn,TCsn)]
836 * Instantiates WF_INDUCTION_THM, getting Sinduct and then tries to prove
837 * recursion induction (Rinduct) by proving the antecedent of Sinduct from
838 * the antecedent of Rinduct.
839 *---------------------------------------------------------------------------*)
840 fun mk_induction thy {fconst, R, SV, pat_TCs_list} =
841 let val tych = Thry.typecheck thy
842 val Sinduction = R.UNDISCH (R.ISPEC (tych R) Thms.WF_INDUCTION_THM)
843 val (pats,TCsl) = ListPair.unzip pat_TCs_list
844 val case_thm = complete_cases thy pats
845 val domain = (type_of o hd) pats
846 val Pname = Name.variant (List.foldr (Library.foldr OldTerm.add_term_names)
848 val P = Free(Pname, domain --> HOLogic.boolT)
849 val Sinduct = R.SPEC (tych P) Sinduction
850 val Sinduct_assumf = S.rand ((#ant o S.dest_imp o concl) Sinduct)
851 val Rassums_TCl' = map (build_ih fconst (P,SV)) pat_TCs_list
852 val (Rassums,TCl') = ListPair.unzip Rassums_TCl'
853 val Rinduct_assum = R.ASSUME (tych (S.list_mk_conj Rassums))
854 val cases = map (fn pat => Term.betapply (Sinduct_assumf, pat)) pats
855 val tasks = U.zip3 cases TCl' (R.CONJUNCTS Rinduct_assum)
856 val proved_cases = map (prove_case fconst thy) tasks
857 val v = Free (Name.variant (List.foldr OldTerm.add_term_names [] (map concl proved_cases))
861 val substs = map (R.SYM o R.ASSUME o tych o (curry HOLogic.mk_eq v)) pats
862 val proved_cases1 = ListPair.map (fn (th,th') => R.SUBS[th]th')
863 (substs, proved_cases)
864 val abs_cases = map (LEFT_ABS_VSTRUCT tych) proved_cases1
865 val dant = R.GEN vtyped (R.DISJ_CASESL (R.ISPEC vtyped case_thm) abs_cases)
866 val dc = R.MP Sinduct dant
867 val Parg_ty = type_of(#Bvar(S.dest_forall(concl dc)))
868 val vars = map (gvvariant[Pname]) (S.strip_prod_type Parg_ty)
869 val dc' = fold_rev (R.GEN o tych) vars
870 (R.SPEC (tych(S.mk_vstruct Parg_ty vars)) dc)
872 R.GEN (tych P) (R.DISCH (tych(concl Rinduct_assum)) dc')
874 handle U.ERR _ => raise TFL_ERR "mk_induction" "failed derivation";
879 (*---------------------------------------------------------------------------
883 *---------------------------------------------------------------------------*)
886 fun simplify_induction thy hth ind =
887 let val tych = Thry.typecheck thy
888 val (asl,_) = R.dest_thm ind
889 val (_,tc_eq_tc') = R.dest_thm hth
890 val tc = S.lhs tc_eq_tc'
893 if (can (Thry.match_term thy asm) tc)
896 (R.MATCH_MP Thms.simp_thm (R.DISCH (tych asm) ind))
903 (*---------------------------------------------------------------------------
904 * The termination condition is an antecedent to the rule, and an
905 * assumption to the theorem.
906 *---------------------------------------------------------------------------*)
907 fun elim_tc tcthm (rule,induction) =
908 (R.MP rule tcthm, R.PROVE_HYP tcthm induction)
912 if !trace then writeln (cat_lines (s :: map Display.string_of_thm L))
915 fun trace_cterms s L =
916 if !trace then writeln (cat_lines (s :: map Display.string_of_cterm L))
920 fun postprocess strict {wf_tac, terminator, simplifier} theory {rules,induction,TCs} =
921 let val tych = Thry.typecheck theory
922 val prove = R.prove strict;
924 (*---------------------------------------------------------------------
925 * Attempt to eliminate WF condition. It's the only assumption of rules
926 *---------------------------------------------------------------------*)
927 val (rules1,induction1) =
928 let val thm = prove(tych(HOLogic.mk_Trueprop
929 (hd(#1(R.dest_thm rules)))),
931 in (R.PROVE_HYP thm rules, R.PROVE_HYP thm induction)
932 end handle U.ERR _ => (rules,induction);
934 (*----------------------------------------------------------------------
935 * The termination condition (tc) is simplified to |- tc = tc' (there
936 * might not be a change!) and then 3 attempts are made:
938 * 1. if |- tc = T, then eliminate it with eqT; otherwise,
939 * 2. apply the terminator to tc'. If |- tc' = T then eliminate; else
940 * 3. replace tc by tc' in both the rules and the induction theorem.
941 *---------------------------------------------------------------------*)
943 fun simplify_tc tc (r,ind) =
944 let val tc1 = tych tc
945 val _ = trace_cterms "TC before simplification: " [tc1]
946 val tc_eq = simplifier tc1
947 val _ = trace_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 = subtract (op aconv) tcs ftcs
994 val extra_tc_thms = map simplify_nested_tc extra_tcs
995 val (r1,ind1) = fold 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}