1 (* Title: Tools/eqsubst.ML
2 Author: Lucas Dixon, University of Edinburgh
4 A proof method to perform a substiution using an equation.
9 (* a type abbreviation for match information *)
11 ((indexname * (sort * typ)) list (* type instantiations *)
12 * (indexname * (typ * term)) list) (* term instantiations *)
13 * (string * typ) list (* fake named type abs env *)
14 * (string * typ) list (* type abs env *)
15 * term (* outer term *)
20 * Zipper.T (* focusterm to search under *)
22 exception eqsubst_occL_exp of
23 string * int list * Thm.thm list * int * Thm.thm
25 (* low level substitution functions *)
26 val apply_subst_in_asm :
30 (Thm.cterm list * int * 'a * Thm.thm) * match -> Thm.thm Seq.seq
31 val apply_subst_in_concl :
34 Thm.cterm list * Thm.thm ->
35 Thm.thm -> match -> Thm.thm Seq.seq
37 (* matching/unification within zippers *)
39 Context.theory -> Term.term -> Zipper.T -> match option
41 Context.theory -> int -> Term.term -> Zipper.T -> match Seq.seq
43 (* skipping things in seq seq's *)
45 (* skipping non-empty sub-sequences but when we reach the end
46 of the seq, remembering how much we have left to skip. *)
47 datatype 'a skipseq = SkipMore of int
48 | SkipSeq of 'a Seq.seq Seq.seq;
50 val skip_first_asm_occs_search :
51 ('a -> 'b -> 'c Seq.seq Seq.seq) ->
52 'a -> int -> 'b -> 'c skipseq
53 val skip_first_occs_search :
54 int -> ('a -> 'b -> 'c Seq.seq Seq.seq) -> 'a -> 'b -> 'c Seq.seq
55 val skipto_skipseq : int -> 'a Seq.seq Seq.seq -> 'a skipseq
60 int list -> Thm.thm list -> int -> Thm.thm -> Thm.thm Seq.seq
61 val eqsubst_asm_tac' :
63 (searchinfo -> int -> Term.term -> match skipseq) ->
64 int -> Thm.thm -> int -> Thm.thm -> Thm.thm Seq.seq
67 int list -> (* list of occurences to rewrite, use [0] for any *)
68 Thm.thm list -> int -> Thm.thm -> Thm.thm Seq.seq
70 Proof.context -> (* proof context *)
71 (searchinfo -> Term.term -> match Seq.seq) (* search function *)
72 -> Thm.thm (* equation theorem to rewrite with *)
73 -> int (* subgoal number in goal theorem *)
74 -> Thm.thm (* goal theorem *)
75 -> Thm.thm Seq.seq (* rewritten goal theorem *)
78 val fakefree_badbounds :
79 (string * Term.typ) list ->
81 (string * Term.typ) list * (string * Term.typ) list * Term.term
84 (Term.term -> Term.term) ->
85 ('a * Term.typ) list -> Term.term -> Term.term
87 (* preparing substitution *)
88 val prep_meta_eq : Proof.context -> Thm.thm -> Thm.thm list
89 val prep_concl_subst :
90 int -> Thm.thm -> (Thm.cterm list * Thm.thm) * searchinfo
91 val prep_subst_in_asm :
92 int -> Thm.thm -> int ->
93 (Thm.cterm list * int * int * Thm.thm) * searchinfo
94 val prep_subst_in_asms :
96 ((Thm.cterm list * int * int * Thm.thm) * searchinfo) list
97 val prep_zipper_match :
98 Zipper.T -> Term.term * ((string * Term.typ) list * (string * Term.typ) list * Term.term)
100 (* search for substitutions *)
101 val valid_match_start : Zipper.T -> bool
102 val search_lr_all : Zipper.T -> Zipper.T Seq.seq
103 val search_lr_valid : (Zipper.T -> bool) -> Zipper.T -> Zipper.T Seq.seq
104 val searchf_lr_unify_all :
105 searchinfo -> Term.term -> match Seq.seq Seq.seq
106 val searchf_lr_unify_valid :
107 searchinfo -> Term.term -> match Seq.seq Seq.seq
108 val searchf_bt_unify_valid :
109 searchinfo -> Term.term -> match Seq.seq Seq.seq
112 val ith_syntax : Args.T list -> int list * Args.T list
113 val options_syntax : Args.T list -> bool * Args.T list
115 (* Isar level hooks *)
116 val eqsubst_asm_meth : Proof.context -> int list -> Thm.thm list -> Proof.method
117 val eqsubst_meth : Proof.context -> int list -> Thm.thm list -> Proof.method
118 val subst_meth : Method.src -> Proof.context -> Proof.method
119 val setup : theory -> theory
127 structure Z = Zipper;
129 (* changes object "=" to meta "==" which prepares a given rewrite rule *)
130 fun prep_meta_eq ctxt =
131 Simplifier.mksimps (Simplifier.local_simpset_of ctxt) #> map Drule.zero_var_indexes;
134 (* a type abriviation for match information *)
136 ((indexname * (sort * typ)) list (* type instantiations *)
137 * (indexname * (typ * term)) list) (* term instantiations *)
138 * (string * typ) list (* fake named type abs env *)
139 * (string * typ) list (* type abs env *)
140 * term (* outer term *)
145 * Zipper.T (* focusterm to search under *)
148 (* skipping non-empty sub-sequences but when we reach the end
149 of the seq, remembering how much we have left to skip. *)
150 datatype 'a skipseq = SkipMore of int
151 | SkipSeq of 'a Seq.seq Seq.seq;
152 (* given a seqseq, skip the first m non-empty seq's, note deficit *)
153 fun skipto_skipseq m s =
159 (case Seq.pull h of NONE => skip_occs n t
160 | SOME _ => if n <= 1 then SkipSeq (Seq.cons h t)
161 else skip_occs (n - 1) t)
162 in (skip_occs m s) end;
164 (* note: outerterm is the taget with the match replaced by a bound
165 variable : ie: "P lhs" beocmes "%x. P x"
166 insts is the types of instantiations of vars in lhs
167 and typinsts is the type instantiations of types in the lhs
168 Note: Final rule is the rule lifted into the ontext of the
170 fun mk_foo_match mkuptermfunc Ts t =
172 val ty = Term.type_of t
173 val bigtype = (rev (map snd Ts)) ---> ty
175 | mk_foo i t = mk_foo (i - 1) (t $ (Bound (i - 1)))
176 val num_of_bnds = (length Ts)
177 (* foo_term = "fooabs y0 ... yn" where y's are local bounds *)
178 val foo_term = mk_foo num_of_bnds (Bound num_of_bnds)
179 in Abs("fooabs", bigtype, mkuptermfunc foo_term) end;
181 (* T is outer bound vars, n is number of locally bound vars *)
182 (* THINK: is order of Ts correct...? or reversed? *)
183 fun fakefree_badbounds Ts t =
184 let val (FakeTs,Ts,newnames) =
185 List.foldr (fn ((n,ty),(FakeTs,Ts,usednames)) =>
186 let val newname = Name.variant usednames n
187 in ((RWTools.mk_fake_bound_name newname,ty)::FakeTs,
189 newname::usednames) end)
192 in (FakeTs, Ts, Term.subst_bounds (map Free FakeTs, t)) end;
194 (* before matching we need to fake the bound vars that are missing an
195 abstraction. In this function we additionally construct the
196 abstraction environment, and an outer context term (with the focus
197 abstracted out) for use in rewriting with RWInst.rw *)
198 fun prep_zipper_match z =
202 val Ts = Z.C.nty_ctxt c
203 val (FakeTs', Ts', t') = fakefree_badbounds Ts t
204 val absterm = mk_foo_match (Z.C.apply c) Ts' t'
206 (t', (FakeTs', Ts', absterm))
209 (* Matching and Unification with exception handled *)
210 fun clean_match thy (a as (pat, t)) =
211 let val (tyenv, tenv) = Pattern.match thy a (Vartab.empty, Vartab.empty)
212 in SOME (Vartab.dest tyenv, Vartab.dest tenv)
213 end handle Pattern.MATCH => NONE;
215 (* given theory, max var index, pat, tgt; returns Seq of instantiations *)
216 fun clean_unify thry ix (a as (pat, tgt)) =
218 (* type info will be re-derived, maybe this can be cached
220 val pat_ty = Term.type_of pat;
221 val tgt_ty = Term.type_of tgt;
222 (* is it OK to ignore the type instantiation info?
223 or should I be using it? *)
225 SOME (Sign.typ_unify thry (pat_ty, tgt_ty) (Vartab.empty, ix))
226 handle Type.TUNIFY => NONE;
229 SOME (typinsttab, ix2) =>
231 (* is it right to throw away the flexes?
232 or should I be using them somehow? *)
234 (Vartab.dest (Envir.type_env env),
236 val initenv = Envir.Envir {asol = Vartab.empty,
237 iTs = typinsttab, maxidx = ix2};
238 val useq = Unify.smash_unifiers thry [a] initenv
239 handle UnequalLengths => Seq.empty
240 | Term.TERM _ => Seq.empty;
241 fun clean_unify' useq () =
242 (case (Seq.pull useq) of
244 | SOME (h,t) => SOME (mk_insts h, Seq.make (clean_unify' t)))
245 handle UnequalLengths => NONE
246 | Term.TERM _ => NONE
248 (Seq.make (clean_unify' useq))
253 (* Matching and Unification for zippers *)
254 (* Note: Ts is a modified version of the original names of the outer
255 bound variables. New names have been introduced to make sure they are
256 unique w.r.t all names in the term and each other. usednames' is
257 oldnames + new names. *)
258 fun clean_match_z thy pat z =
259 let val (t, (FakeTs,Ts,absterm)) = prep_zipper_match z in
260 case clean_match thy (pat, t) of
262 | SOME insts => SOME (insts, FakeTs, Ts, absterm) end;
263 (* ix = max var index *)
264 fun clean_unify_z sgn ix pat z =
265 let val (t, (FakeTs, Ts,absterm)) = prep_zipper_match z in
266 Seq.map (fn insts => (insts, FakeTs, Ts, absterm))
267 (clean_unify sgn ix (t, pat)) end;
271 type trace_subst_errT = int (* subgoal *)
272 * thm (* thm with all goals *)
273 * (Thm.cterm list (* certified free var placeholders for vars *)
274 * thm) (* trivial thm of goal concl *)
275 (* possible matches/unifiers *)
277 * (((indexname * typ) list (* type instantiations *)
278 * (indexname * term) list ) (* term instantiations *)
279 * (string * typ) list (* Type abs env *)
280 * term) (* outer term *);
282 val trace_subst_err = (ref NONE : trace_subst_errT option ref);
283 val trace_subst_search = ref false;
284 exception trace_subst_exp of trace_subst_errT;
288 fun bot_left_leaf_of (l $ r) = bot_left_leaf_of l
289 | bot_left_leaf_of (Abs(s,ty,t)) = bot_left_leaf_of t
290 | bot_left_leaf_of x = x;
292 (* Avoid considering replacing terms which have a var at the head as
293 they always succeed trivially, and uninterestingly. *)
294 fun valid_match_start z =
295 (case bot_left_leaf_of (Z.trm z) of
299 (* search from top, left to right, then down *)
300 val search_lr_all = ZipperSearch.all_bl_ur;
302 (* search from top, left to right, then down *)
303 fun search_lr_valid validf =
305 fun sf_valid_td_lr z =
306 let val here = if validf z then [Z.Here z] else [] in
308 of _ $ _ => [Z.LookIn (Z.move_down_left z)]
310 @ [Z.LookIn (Z.move_down_right z)]
311 | Abs _ => here @ [Z.LookIn (Z.move_down_abs z)]
314 in Z.lzy_search sf_valid_td_lr end;
316 (* search from bottom to top, left to right *)
318 fun search_bt_valid validf =
320 fun sf_valid_td_lr z =
321 let val here = if validf z then [Z.Here z] else [] in
323 of _ $ _ => [Z.LookIn (Z.move_down_left z),
324 Z.LookIn (Z.move_down_right z)] @ here
325 | Abs _ => [Z.LookIn (Z.move_down_abs z)] @ here
328 in Z.lzy_search sf_valid_td_lr end;
330 fun searchf_unify_gen f (sgn, maxidx, z) lhs =
331 Seq.map (clean_unify_z sgn maxidx lhs)
334 (* search all unifications *)
335 val searchf_lr_unify_all =
336 searchf_unify_gen search_lr_all;
338 (* search only for 'valid' unifiers (non abs subterms and non vars) *)
339 val searchf_lr_unify_valid =
340 searchf_unify_gen (search_lr_valid valid_match_start);
342 val searchf_bt_unify_valid =
343 searchf_unify_gen (search_bt_valid valid_match_start);
345 (* apply a substitution in the conclusion of the theorem th *)
346 (* cfvs are certified free var placeholders for goal params *)
347 (* conclthm is a theorem of for just the conclusion *)
348 (* m is instantiation/match information *)
349 (* rule is the equation for substitution *)
350 fun apply_subst_in_concl i th (cfvs, conclthm) rule m =
351 (RWInst.rw m rule conclthm)
352 |> IsaND.unfix_frees cfvs
353 |> RWInst.beta_eta_contract
354 |> (fn r => Tactic.rtac r i th);
356 (* substitute within the conclusion of goal i of gth, using a meta
357 equation rule. Note that we assume rule has var indicies zero'd *)
358 fun prep_concl_subst i gth =
360 val th = Thm.incr_indexes 1 gth;
361 val tgt_term = Thm.prop_of th;
363 val sgn = Thm.theory_of_thm th;
364 val ctermify = Thm.cterm_of sgn;
365 val trivify = Thm.trivial o ctermify;
367 val (fixedbody, fvs) = IsaND.fix_alls_term i tgt_term;
368 val cfvs = rev (map ctermify fvs);
370 val conclterm = Logic.strip_imp_concl fixedbody;
371 val conclthm = trivify conclterm;
372 val maxidx = Thm.maxidx_of th;
373 val ft = ((Z.move_down_right (* ==> *)
374 o Z.move_down_left (* Trueprop *)
376 o Thm.prop_of) conclthm)
378 ((cfvs, conclthm), (sgn, maxidx, ft))
381 (* substitute using an object or meta level equality *)
382 fun eqsubst_tac' ctxt searchf instepthm i th =
384 val (cvfsconclthm, searchinfo) = prep_concl_subst i th;
385 val stepthms = Seq.of_list (prep_meta_eq ctxt instepthm);
386 fun rewrite_with_thm r =
387 let val (lhs,_) = Logic.dest_equals (Thm.concl_of r);
388 in searchf searchinfo lhs
389 |> Seq.maps (apply_subst_in_concl i th cvfsconclthm r) end;
390 in stepthms |> Seq.maps rewrite_with_thm end;
393 (* distinct subgoals *)
394 fun distinct_subgoals th =
395 the_default th (SINGLE distinct_subgoals_tac th);
397 (* General substitution of multiple occurances using one of
401 exception eqsubst_occL_exp of
402 string * (int list) * (thm list) * int * thm;
403 fun skip_first_occs_search occ srchf sinfo lhs =
404 case (skipto_skipseq occ (srchf sinfo lhs)) of
405 SkipMore _ => Seq.empty
406 | SkipSeq ss => Seq.flat ss;
408 (* The occL is a list of integers indicating which occurence
409 w.r.t. the search order, to rewrite. Backtracking will also find later
410 occurences, but all earlier ones are skipped. Thus you can use [0] to
411 just find all rewrites. *)
413 fun eqsubst_tac ctxt occL thms i th =
414 let val nprems = Thm.nprems_of th in
415 if nprems < i then Seq.empty else
416 let val thmseq = (Seq.of_list thms)
417 fun apply_occ occ th =
419 (fn r => eqsubst_tac'
421 (skip_first_occs_search
422 occ searchf_lr_unify_valid) r
423 (i + ((Thm.nprems_of th) - nprems))
426 Library.sort (Library.rev_order o Library.int_ord) occL;
428 Seq.map distinct_subgoals (Seq.EVERY (map apply_occ sortedoccL) th)
431 handle THM _ => raise eqsubst_occL_exp ("THM",occL,thms,i,th);
434 (* inthms are the given arguments in Isar, and treated as eqstep with
435 the first one, then the second etc *)
436 fun eqsubst_meth ctxt occL inthms =
437 SIMPLE_METHOD' (eqsubst_tac ctxt occL inthms);
439 (* apply a substitution inside assumption j, keeps asm in the same place *)
440 fun apply_subst_in_asm i th rule ((cfvs, j, ngoalprems, pth),m) =
442 val th2 = Thm.rotate_rule (j - 1) i th; (* put premice first *)
444 (RWInst.rw m rule pth)
445 |> (Seq.hd o prune_params_tac)
446 |> Thm.permute_prems 0 ~1 (* put old asm first *)
447 |> IsaND.unfix_frees cfvs (* unfix any global params *)
448 |> RWInst.beta_eta_contract; (* normal form *)
451 |> Tactic.make_elim (* make into elim rule *)
452 |> Thm.lift_rule (th2, i); (* lift into context *)
455 (* ~j because new asm starts at back, thus we subtract 1 *)
456 Seq.map (Thm.rotate_rule (~j) ((Thm.nprems_of rule) + i))
457 (Tactic.dtac preelimrule i th2)
460 false (* use unification *)
461 (true, (* elim resolution *)
462 elimrule, (2 + (Thm.nprems_of rule)) - ngoalprems)
467 (* prepare to substitute within the j'th premise of subgoal i of gth,
468 using a meta-level equation. Note that we assume rule has var indicies
469 zero'd. Note that we also assume that premt is the j'th premice of
470 subgoal i of gth. Note the repetition of work done for each
471 assumption, i.e. this can be made more efficient for search over
472 multiple assumptions. *)
473 fun prep_subst_in_asm i gth j =
475 val th = Thm.incr_indexes 1 gth;
476 val tgt_term = Thm.prop_of th;
478 val sgn = Thm.theory_of_thm th;
479 val ctermify = Thm.cterm_of sgn;
480 val trivify = Thm.trivial o ctermify;
482 val (fixedbody, fvs) = IsaND.fix_alls_term i tgt_term;
483 val cfvs = rev (map ctermify fvs);
485 val asmt = nth (Logic.strip_imp_prems fixedbody) (j - 1);
486 val asm_nprems = length (Logic.strip_imp_prems asmt);
488 val pth = trivify asmt;
489 val maxidx = Thm.maxidx_of th;
491 val ft = ((Z.move_down_right (* trueprop *)
494 in ((cfvs, j, asm_nprems, pth), (sgn, maxidx, ft)) end;
496 (* prepare subst in every possible assumption *)
497 fun prep_subst_in_asms i gth =
498 map (prep_subst_in_asm i gth)
499 ((fn l => Library.upto (1, length l))
500 (Logic.prems_of_goal (Thm.prop_of gth) i));
503 (* substitute in an assumption using an object or meta level equality *)
504 fun eqsubst_asm_tac' ctxt searchf skipocc instepthm i th =
506 val asmpreps = prep_subst_in_asms i th;
507 val stepthms = Seq.of_list (prep_meta_eq ctxt instepthm);
508 fun rewrite_with_thm r =
509 let val (lhs,_) = Logic.dest_equals (Thm.concl_of r)
510 fun occ_search occ [] = Seq.empty
511 | occ_search occ ((asminfo, searchinfo)::moreasms) =
512 (case searchf searchinfo occ lhs of
513 SkipMore i => occ_search i moreasms
515 Seq.append (Seq.map (Library.pair asminfo) (Seq.flat ss))
516 (occ_search 1 moreasms))
517 (* find later substs also *)
519 occ_search skipocc asmpreps |> Seq.maps (apply_subst_in_asm i th r)
521 in stepthms |> Seq.maps rewrite_with_thm end;
524 fun skip_first_asm_occs_search searchf sinfo occ lhs =
525 skipto_skipseq occ (searchf sinfo lhs);
527 fun eqsubst_asm_tac ctxt occL thms i th =
528 let val nprems = Thm.nprems_of th
530 if nprems < i then Seq.empty else
531 let val thmseq = (Seq.of_list thms)
532 fun apply_occ occK th =
535 eqsubst_asm_tac' ctxt (skip_first_asm_occs_search
536 searchf_lr_unify_valid) occK r
537 (i + ((Thm.nprems_of th) - nprems))
540 Library.sort (Library.rev_order o Library.int_ord) occL
542 Seq.map distinct_subgoals
543 (Seq.EVERY (map apply_occ sortedoccs) th)
546 handle THM _ => raise eqsubst_occL_exp ("THM",occL,thms,i,th);
548 (* inthms are the given arguments in Isar, and treated as eqstep with
549 the first one, then the second etc *)
550 fun eqsubst_asm_meth ctxt occL inthms =
551 SIMPLE_METHOD' (eqsubst_asm_tac ctxt occL inthms);
553 (* syntax for options, given "(asm)" will give back true, without
556 (Args.parens (Args.$$$ "asm") >> (K true)) ||
557 (Scan.succeed false);
560 Scan.optional (Args.parens (Scan.repeat OuterParse.nat)) [0];
562 (* combination method that takes a flag (true indicates that subst
563 should be done to an assumption, false = apply to the conclusion of
564 the goal) as well as the theorems to use *)
566 Method.syntax ((Scan.lift options_syntax) -- (Scan.lift ith_syntax) -- Attrib.thms) src
567 #> (fn (((asmflag, occL), inthms), ctxt) =>
568 (if asmflag then eqsubst_asm_meth else eqsubst_meth) ctxt occL inthms);
572 Method.add_method ("subst", subst_meth, "single-step substitution");