1 (* Title: todo's for isac core
2 Author: Walther Neuper 111013
3 (c) copyright due to lincense terms.
6 imports "~~/src/Doc/Prog_Prove/LaTeXsugar"
10 Legend: here + code since 200120
11 \<open>(*DOC\<close> identifies stuff dedicated for isac/Doc/Lucas_Interpreter
12 \<open>(*?!?\<close> identifies a design question
14 Distillation of "rounds of reforms":
15 \<open>TODOs from current changesets\<close> \<rightarrow> \<open>Postponed from current changeset\<close> \<rightarrow>
16 (*to be removed..*) \<rightarrow> \<open>Separated tasks\<close> (*.. to be removed*)
17 \<rightarrow> subsection of \<open>Major reforms\<close>
20 section \<open>TODOs from current changesets\<close>
22 Shift more complicated issues from \<open>Current changeset\<close> to \<open>Postponed from current changeset\<close>
24 subsection \<open>Current changeset\<close>
29 \item LI.do_next (*TODO RM..*)
31 \item RENAME ARGS ContextC.subpbl_to_caller sub_ctxt prog_res ctxt_parent
33 \item RM asm FROM Check_Postcond' (pI, (prog_res, asm)) ... asms are handled by ctxt !
35 \item generate.sml: RM datatype edit = EdUndef | Write | Protect
37 \item ad ERROR @unknown fact all_left in Test_Isac: search "Undefined"
40 \item TermC.list2isalist: typ -> term list -> term .. should NOT requires typ
42 \item restore clarity of "trace_rewrite": "=====" seems to be lost:
43 ##### try calc: HOL.eq' <<<------------------------------- here the error comes from
44 ===== calc. to: ~ False <<<------------------------------- \<not> x = 0 is NOT False
45 ##### try calc: HOL.eq'
46 ##### try thm: not_true
47 ##### try thm: not_false
48 ===== rewrites to: True <<<------------------------------- so x \<noteq> 0 is NOT True
50 \item get_ctxt_LI should replace get_ctxt
52 \item change separate test ----------------------vvv
53 | by_tactic (tac as Tactic.Check_Postcond' _) ptp =
54 LI.by_tactic tac (Istate.e_istate, ContextC.e_ctxt) ptp
55 -----------------^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
57 \item mv Ctree.get_assumptions_ -> Ctree.ctxt_assumptions
59 \item Error_Pattern.check_error_pattern -> Error_Pattern.check
61 \item DEL double code: lev_on +??? in ctree-basic.sml + position
63 \item DEL double code: nxt_specify_init_calc IN specify.sml + step-specify.sml
65 \item ALL CODE: rename spec(ification) --> know(ledge), in Specification: Relation -> Knowledge
67 \item replace ContextC.insert_assumptions by Tactic.insert_assumptions
68 NOTE: DONE in associate, ??missing in LI.do_step ??
70 \item associate: drop ctxt from arg.+ return
72 \item in check_leaf SEPARATE tracing
73 collect all trace_script --> Trace_LI
74 trace_script: replace ' by " in writeln
78 \item NotAss -> Not_Associated | Ass -> Associated
79 \item ContextC.subpbl_to_caller -> subpbl_to_caller
80 \item Tactic.is_empty_tac -> Tactic.is_empty
81 \item Istate.istate2str -> Istate.string_of
83 \item rename Base_Tool.thy <--- Base_Tools
84 \item adopt naming conventions in Knowledge: EqSystem --> Equation_System, etc
86 \item rename field scr in meth
91 \item relocations + renamings
93 \item Rule.terms2str -> TermC.s_to_string
94 Tactic_Def.tac2str -> Tactic_Def.input_to_string
101 \item cleanup test-files
103 \item test/.. tools.sml, atools.sml, ...
110 \item see in Test_Some/300- (*TODO: DOUBLE subst, DROP ONE*)
111 \item ? improve sig. eval_leaf E a v t -> (E, (a, v)) t
114 \item clarify Tactic.Subproblem (domID, pblID) as term in Pstate {act_arg, ...}
115 there it is Free ("Subproblem", "char list \<times> ..
116 instead of Const (|???.Subproblem", "char list \<times> ..
117 AND THE STRING REPRESENTATION IS STRANGE:
118 Subproblem (''Test'',
119 ??.\ <^const> String.char.Char ''LINEAR'' ''univariate'' ''equation''
123 term2str; (*defined by..*)
124 fun term_to_string''' thy t =
126 val ctxt' = Config.put show_markup false (Proof_Context.init_global thy)
127 in Print_Mode.setmp [] (Syntax.string_of_term ctxt') t end;
129 val t = @{term "aaa + bbb"}
130 val t = @{term "Subproblem (''Test'', [''LINEAR'', ''univariate'', ''equation''])"};
132 val sss = Print_Mode.setmp [] (Syntax.string_of_term @{context}) t
134 writeln sss (*.. here perfect: Subproblem (''Test'', [''LINEAR'', ''univariate'', ''equation'']) *)
139 \item LI.locate_input_term cleanup, case ONLY
141 \item revert 9ef6e9e88178 (separate Tactic / Tactic_Def),
142 because tactic.sml after ctree.sml doesn't gain much !
143 ?Tactic.input -> Calc.tactic ?
146 \item collect from_pblobj_or_detail*,
147 -> Rule.rls(*..\<in> ist \<rightarrow> REMOVE*) * (Istate.T * Proof.context) * Program.T
148 \item init_istate ?-> Detail_Rls?
150 \item rename Ctree.pos' --> Pos.pos', type decl duplicate? delete, separate!
152 \item Istate_Def.e_istate -> Istate.empty
156 \item distribute test/../scrtools.sml: WAIT FOR FINAL CLEANUP OF src/../Interpret
161 subsection \<open>Postponed from current changeset\<close>
166 \item cleanup fun me:
167 fun me (*(_, Empty_Tac) p _ _ = raise ERROR ("me: Empty_Tac at " ^ pos'2str p)
168 | me*) (_, tac) p _(*NEW remove*) pt =
169 + -------------------------^^^^^^
170 # see test-code.sml fun me_trace
171 use this also for me, not only for me_ist_ctxt; del. me
172 this requires much updating in all test/*
175 \item generate, generate1: NO thy as arg.
176 Generate.generate1 thy is redundant: is the same during pbl, thus lookup spec
177 redesign return value (originally for output by fun me?)
179 \item shift tests into NEW model.sml (upd, upds_envv, ..)
182 \item clarify handling of contexts ctxt ContextC
185 \item Specify/ works with thy | Interpret/ works with ctxt | MathEngine.step works with ?!?ctxt
187 \item Check_Elementwise "Assumptions": prerequisite for ^^goal
188 rm tactic Check_elementwise "Assumptions" in a way, which keeps it for Minisubpbl
189 rm Assumptions :: bool (* TODO: remove with making ^^^ idle *)
195 \item complete mstools.sml (* survey on handling contexts:
200 \item librarys.ml --> libraryC.sml + text from termC.sml
207 \item concentrate "insert_assumptions" for locate_input_tactic in "associate", ?OR? Tactic.insert_assumptions
208 DONE for find_next_step by Tactic.insert_assumptions m' ctxt
210 \item rm from "generate1" ("Detail_Set_Inst'", Tactic.Detail_Set' ?)
211 \item shift from "applicable_in..Apply_Method" to ? ? ? (is ONLY use-case in appl.sml))
212 \item ?"insert_assumptions" necessary in "init_pstate" ?+++? in "applicable_in" ?+++? "associate"
215 \item DO DELETIONS AFTER analogous concentrations in find_next_step
218 \item ? what is the difference headline <--> cascmd in
219 Subproblem' (spec, oris, headline, fmz_, context, cascmd)
221 \item inform: TermC.parse (Celem.assoc_thy "Isac_Knowledge") istr --> parseNEW context istr
223 \item unify/clarify stac2tac_ |
226 \item extract common code from associate.. stac2tac_xxx
227 \item rename LItool.tac_from_prog -> Tactic.from_prog_tac ? Solve_Tac.from_prog_tac,
232 \item unify in signature LANGUAGE_TOOLS =\\
233 val pblterm: Rule.domID -> Celem.pblID -> term vvv vvv\\
234 val subpbl: string -> string list -> term unify with ^^^
236 \item Telem.safe is questionable: has it been replaced by Safe_Step, Not_Derivable, Helpless, etc?
237 Note: replacement of Istate.safe by Istate.appy_ didn't care much about Telem.safe.
238 If Telem.safe is kept, consider merge with CTbasic.ostate
240 \item remove find_next_step from solve Apply_Method';
241 this enforces Pos.at_first_tactic, which should be dropped, too.
247 subsection \<open>Postponed -?-> separated task ?\<close>
252 \item replace all Ctree.update_* with Ctree.cupdate_problem
254 \item rename (ist as {eval, ...}) -> (ist as {eval_rls, ...})
256 \item exception PROG analogous to TERM
258 \item sig/struc ..elems --> ..elem
260 \item distille CAS-command, CAScmd, etc into a struct
262 \item check location of files:
263 test/Tools/isac/Interpret/ptyps.thy
264 test/Tools/isac/Specify.ptyps.sml
266 \item check occurences of Atools in src/ test/
267 \item Const ("Atools.pow", _) ---> Const ("Base_Tool.pow", _)
271 \item Diff.thy: differentiateX --> differentiate after removal of script-constant
272 \item Test.thy: met_test_sqrt2: deleted?!
274 \item Rule.rew_ord' := overwritel (! Rule.rew_ord', (*<<<---- use KEStore.xxx, too*)
276 \item automatically extrac rls from program-code
277 ? take ["SignalProcessing", "Z_Transform", "Inverse_sub"] as an example ?
279 \item finish output of trace_script with Check_Postcond (useful for SubProblem)
291 section \<open>Separated tasks\<close>
293 This section shall be removed, because
294 the distinction \<open>Simple\<close> | \<open>Simple but laborous\<close> | \<open>Questionable\<close>
295 is too distracting when working on \<open>TODOs from current changesets\<close>.
298 section \<open>Major reforms\<close>
301 subsection \<open>Exception Size raised\<close>
303 During update Isabelle2018 --> Isabelle2019 we noticed, that
304 "isabelle build" uses resources more efficiently than "isabelle jedit".
305 The former works, but the latter causes
307 \item "Exception- Size raised"
309 \item "exception Size raised (line 169 of "./basis/LibrarySupport.sml")"
310 in test/../biegelinie-*.xml.
312 This has been detected after changeset (30cd47104ad7) "lucin: reorganise theories in ProgLang".
314 Find tools to investigate the Exception, and find ways around it eventually.
316 subsection \<open>Cleanup & review signatures wrt. implementation.pdf canonical argument order\<close>
319 \item there are comments in several signatures
320 \item ML_file "~~/src/Tools/isac/Interpret/specification-elems.sml" can be (almost) deleted
321 \item src/../Frontend/: signatures missing
325 subsection \<open>overall structure of code\<close>
328 \item try to separate Isac_Knowledge from MathEngine
329 common base: Knowledge_Author / ..??
331 \item ML_file "~~/src/Tools/isac/Interpret/ctree.sml" (*shift to base in common with Interpret*)
337 subsection \<open>Separate MathEngineBasic/ Specify/ Interpret/ MathEngine/\<close>
342 \item re-organise code for Interpret
344 \item Step*: Step_Specify | Step_Solve | Step
346 \item *.check | *.add ARE TOO LATE IN BUILD with Step_Specify | Step_Solve
347 Specify_Step.check | Specify_Step.add <-- Applicable.applicable_in
348 Solve_Step.check | Solve_Step.add <-- Generate.generate1
352 \item Prog_Tac: fun get_first takes both Prog_Tac + Program --- wait for separate Tactical
353 then shift into common descendant
358 \item ??????????? WHY CAN LI.by_tactic NOT BE REPLACED BY Step_Solve.by_tactic ???????????
363 subsection \<open>Review modelling- + specification-phase\<close>
369 \item check match between args of partial_function and model-pattern of meth;
370 provide error message.
372 \item "--- hack for funpack: generalise handling of meths which extend problem items ---"
374 \item see several locations of hack in code
375 \item these locations are NOT sufficient, see
376 test/../biegelinie-3.sml --- IntegrierenUndKonstanteBestimmen2: Bsp.7.70. auto ---
377 \item "fun associate" "match_ags ..dI" instead "..pI" breaks many tests, however,
378 this would be according to survey Notes (3) in src/../calchead.sml.
380 \item see "failed trial to generalise handling of meths"98298342fb6d
381 \item abstract specify + nxt_specif to common aux-funs;
382 see e.g. "--- hack for funpack: generalise handling of meths which extend problem items ---"
384 \item type model = itm list ?
385 \item review survey Notes in src/../calchead.sml: they are questionable
386 \item review copy-named, probably two issues commingled
388 \item special handling of "#Find#, because it is not a formal argument of partial_function
389 \item special naming for solutions of equation solving: x_1, x_2, ...
392 \item structure Tactic Specify -?-> Proglang (would require Model., Selem.)
394 \item this has been written in one go:
396 \item reconsidering Model.max_vt, use problem with meth ["DiffApp","max_by_calculus"]
397 \item reconsider add_field': where is it used for what? Shift into mk_oris
398 \item reconsider match_itms_oris: where is it used for what? max_vt ONLY???
399 \item in Specify_Method search root-oris for items (e.g. "errorBound"), #1# in survey
400 \item Specify_Problem, Specify_Method: check respective identifiers after re-Specify_
401 (relevant for pre-condition)
402 \item unify match_ags to mk_oris1..N with different args (fmz | pat list, pbl | meth
407 subsection \<open>taci list, type step\<close>
409 taci was, most likely, invented to make "fun me" more efficient by avoiding duplicate rewrite,
410 and probably, because the Kernel interface separated setNextTactic and applyTactic.
411 Both are no good reasons to make code more complicated.
413 !!! taci list is used in do_next !!!
417 \item can lev_on_total replace lev_on ? ..Test_Isac_Short + rename lev_on_total -> lev_on
419 \item Step* functions should return Calc.T instead of Chead.calcstate'
421 \item states.sml: check, when "length tacis > 1"
422 \item in Test_Isac.thy there is only 1 error in Interpret/inform.sml
423 \item (*WN190713 REMOVE: "creating a new node" was never implemented for more than one node?!?
425 \item brute force setting all empty ([], [], ptp) works!?! but ptp causes errors -- investigate!
429 subsection \<open>Ctree\<close>
432 # mixture pos' .. pos in cappend_*, append_* is confusing
433 # existpt p pt andalso Tactic.is_empty_tac DIFFERENT IN append_*, cappend_* is confusing
434 "exception PTREE "get_obj: pos =" ^^^^^: ^^^^ due to cut !!!
435 NOTE: exn IN if..andalso.. IS NOT!!! DETECTED, THIS is confusing
436 see test/../--- Minisubpbl/800-append-on-Frm.sml ---
437 # ?!? "cut branches below cannot be decided here" in append_atomic
438 # sign. of functions too different ?!?canonical arg.order ?!?
441 \item remove update_branch, update_*? -- make branch, etc args of append_*
443 \item close sig Ctree, contains cappend_* ?only? --- ?make parallel to ?Pide_Store?
445 \item unify args to Ctree.state (pt, p)
446 \item fun update_env .. repl_env \<rightarrow>update_istate
453 subsection \<open>replace theory/thy by context/ctxt\<close>
457 \item Specify/ works with thy | Interpret/ works with ctxt | MathEngine.step works with ?!?ctxt
458 special case: Tactic.Refine_Problem
460 \item theory can be retrieved from ctxt by Proof_Context.theory_of
462 \item cleaup the many conversions string -- theory
463 \item make dest_spec --> (theory, pblID, metID) ?+ common_subthy ?
464 \item 1. Rewrite.eval_true_, then
465 LItool.check_leaf, Rewrite.eval_prog_expr, Generate.generate1, LItool.tac_from_prog.
467 let val thy = Celem.assoc_thy "Isac_Knowledge";(*TODO*)
470 \item in locate_input_tactic .. ?scan_dn1?; Program.is_eval_expr .use Term.exists_Const
471 \item locate_input_tactic: get_simplifier cstate (*TODO: shift to init_istate*)
472 \item push srls into pstate
473 \item lucas-intrpreter.locate_input_tactic: scan_to_tactic1 srls tac cstate (progr, Rule.e_rls)
478 subsection \<open>Rfuns, Begin_/End_Detail', Rrls, Istate\<close>
480 remove Rfuns -> Rule.Prog, Rule.EmptyScr
481 consider separating spec.funs. to ?Inter_Steps?
483 \item ^^^+ see from_pblobj_or_detail_thm, from_pblobj_or_detail_calc, ...
485 \item probably only "normal_form" seems to be needed
486 \item deleted Rfuns in NEW "locate_input_tactic": no active test for "locate_rule"
487 but that seems desirable
488 \item ?how is the relation to reverse-rewriting ???
489 \item "Rfuns" markers in test/../rational
491 \item datatype istate (Istate.T): remove RrlsState, pstate: use Rrls only for creating results beyond
492 rewriting and/or respective intermediate steps (e.g. cancellation of fractions).
493 Thus we get a 1-step-action which does NOT require a state beyond istate/pstate.
494 Thus we drastically reduce complexity, also get rid of "fun from_pblobj_or_detail_calc" , etc.
495 \item debug ^^^ related: is there an occurence of Steps with more than 1 element?
497 \item and do_next (* WN1907: ?only for Begin_/End_Detail' DEL!!!*)
499 \item shouldn't go Rfuns from Rewrite --> Rewrite_Set; they behave similar to "fun interSteps" ?
502 subsection \<open>Inverse_Z_Transform.thy\<close>
505 \item\label{new-var-rhs} rule1..6, ruleZY introduce new variables on the rhs of the rewrite-rule.
506 ? replace rewriting with substitution ?!?
507 The problem is related to the decision of typing for "d_d" and making bound variables free (while
508 shifting specific handling in equation solving etc. to the meta-logic).
509 \item Find "stepResponse (x[n::real]::bool)" is superfluous, because immediately used by
510 rewrite-rules; see \ref{new-var-rhs}.
511 \item Reconsider whole problem:
512 input only the polynomial as argument of partial_function, in ([1], Frm) compile lhs "X z" ?
515 subsection \<open>Adopt Isabelle's numerals for Isac\<close>
518 \item replace numerals of type "real" by "nat" in some specific functions from ListC.thy
519 and have both representations in parallel for "nat".
524 subsection \<open>Redesign equation solver\<close>
526 Existing solver is structured along the WRONG assumption,
527 that Poly.thy must be the LAST thy among all thys involved -- while the opposite is the case.
529 Preliminary solution: all inappropriately located thms are collected in Base_Tools.thy
531 subsection \<open>Finetunig required for xmldata in kbase\<close>
533 See xmldata https://intra.ist.tugraz.at/hg/isac/rev/5b222a649390
534 and in kbase html-representation generated from these xmldata.
535 Notes in ~~/xmldata/TODO.txt.
538 section \<open>Hints for further development\<close>
541 subsection \<open>Coding standards & some explanations for math-authors\<close>
542 text \<open>copy from doc/math-eng.tex WN.28.3.03
543 WN071228 extended\<close>
545 subsubsection \<open>Identifiers\<close>
546 text \<open>Naming is particularily crucial, because Isabelles name space is global, and isac does
547 not yet use the novel locale features introduces by Isar. For instance, {\tt probe} sounds
548 reasonable as (1) a description in the model of a problem-pattern, (2) as an element of the
549 problem hierarchies key, (3) as a socalled CAS-command, (4) as the name of a related script etc.
550 However, all the cases (1)..(4) require different typing for one and the same
551 identifier {\tt probe} which is impossible, and actually leads to strange errors
552 (for instance (1) is used as string, except in a script addressing a Subproblem).
554 These are the preliminary rules for naming identifiers>
556 \item [elements of a key] into the hierarchy of problems or methods must not contain
557 capital letters and may contain underscrores, e.g. {\tt probe, for_polynomials}.
558 \item [descriptions in problem-patterns] must contain at least 1 capital letter and
559 must not contain underscores, e.g. {\tt Probe, forPolynomials}.
560 \item [CAS-commands] follow the same rules as descriptions in problem-patterns above, thus
561 beware of conflicts~!
562 \item [script identifiers] always end with {\tt Program}, e.g. {\tt ProbeScript}.
566 %WN071228 extended\<close>
568 subsubsection \<open>Rule sets\<close>
569 text \<open>The actual version of the coding standards for rulesets is in {\tt /IsacKnowledge/Atools.ML
570 where it can be viewed using the knowledge browsers.
572 There are rulesets visible to the student, and there are rulesets visible (in general) only for
573 math authors. There are also rulesets which {\em must} exist for {\em each} theory;
574 these contain the identifier of the respective theory (including all capital letters)
575 as indicated by {\it Thy} below.
578 \item [norm\_{\it Thy}] exists for each theory, and {\em efficiently} calculates a
579 normalform for all terms which can be expressed by the definitions of the respective theory
580 (and the respective parents).
581 \item [simplify\_{\it Thy}] exists for each theory, and calculates a normalform for all terms
582 which can be expressed by the definitions of the respective theory (and the respective parents)
583 such, that the rewrites can be presented to the student.
584 \item [calculate\_{\it Thy}] exists for each theory, and evaluates terms with
585 numerical constants only (i.e. all terms which can be expressed by the definitions of
586 the respective theory and the respective parent theories). In particular, this ruleset includes
587 evaluating in/equalities with numerical constants only.
588 WN.3.7.03: may be dropped due to more generality: numericals and non-numericals
589 are logically equivalent, where the latter often add to the assumptions
590 (e.g. in Check_elementwise).
593 The above rulesets are all visible to the user, and also may be input;
594 thus they must be contained in {\tt Theory_Data} (KEStore_Elems.add_rlss,
595 KEStore_Elems.get_rlss). All these rulesets must undergo a preparation
596 using the function {\tt prep_rls'}, which generates a script for stepwise rewriting etc.
597 The following rulesets are used for internal purposes and usually invisible to the (naive) user:
604 {\tt Rule.append_rls, Rule.merge_rls, remove_rls} TODO
607 subsection \<open>get proof-state\<close>
609 Re: [isabelle] Programatically get "this"
610 ----------------------------------------------------
611 So here is my (Makarius') version of your initial example, following these principles:
617 val ctxt = @{context};
620 Name_Space.full_name (Proof_Context.naming_of ctxt) (Binding.name Auto_Bind.thisN);
621 val this = #thms (the (Proof_Context.lookup_fact ctxt this_name));
626 subsection \<open>write Specification to jEdit\<close>
628 Re: [isabelle] Printing terms with type annotations
629 ----------------------------------------------------
630 On 06/02/2019 17:52, Moa Johansson wrote:
632 > I’m writing some code that should create a snippet of Isar script.
634 This is how Sledgehammer approximates this:
636 http://isabelle.in.tum.de/repos/isabelle/file/Isabelle2018/src/HOL/Tools/Sledgehammer/sledgehammer_isar_proof.ML#l299
638 (The module name already shows that the proper terminology is "Isar
639 proof" (or "Isar proof text"). Proof scripts are a thing from the past,
640 before Isar. You can emulate old-style proof scripts via a sequence of
641 'apply' commands, but this is improper Isar.)
643 Note that there is no standard function in Isabelle/Pure, because the
644 problem to print just the right amount of type information is very
645 complex and not fully solved. One day, after 1 or 2 rounds of
646 refinements over the above approach, it might become generally available.
648 subsection \<open>follow Isabelle conventions (*Does not yet work in Isabelle2018\<close>
650 isabelle update -u path_cartouches
651 isabelle update -u inner_syntax_cartouches
653 section \<open>Questions to Isabelle experts\<close>
658 \item ?OK Test_Isac_Short with
659 LI.by_tactic tac (get_istate_LI pt p, get_ctxt_LI pt p) ptp
661 LI.by_tactic tac (Istate.e_istate, ContextC.e_ctxt) ptp
664 \item test from last CS with outcommented re-def of code ->
665 -> \<open>further tests additional to src/.. files\<close>
666 ADDTESTS/redefined-code.sml
668 \item efb749b79361 Test_Some_Short.thy has 2 errors, which disappear in thy ?!?:
669 ML_file "Interpret/error-pattern.sml" Undefined fact: "all_left"
670 ML_file "Knowledge/inssort.sml" Undefined fact: "xfoldr_Nil"
672 \item what is the actual replacement of "hg log --follow" ?
674 \item how HANDLE these exceptions, e.g.:
675 Syntax.read_term ctxt "Randbedingungen y 0 = (0::real), y L = 0, M_b 0 = 0, M_b L = 0]"
678 Failed to parse term"
680 \item how cope with "exception Size raised (line 171 of "./basis/LibrarySupport.sml")"
681 e.g. in test/Interpret/lucas-interpreter.sml
687 section \<open>For copy & paste\<close>
705 subsection \<open>xxx\<close>
706 subsubsection \<open>xxx\<close>