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19 <img class="icon32" alt="[\lambda\delta home]" title="\lambda\delta home" src="http://lambdadelta.info/images/crux_32.png" />
22 <div class="head1">cic:/matita/lambdadelta/basic_2/ (core λδ version 2)</div>
24 <img class="rule" alt="[Spacer]" title="\lambda\delta rainbow rule" src="http://lambdadelta.info/images/rainbow.png" />
26 <div xmlns:ld="http://lambdadelta.info/" class="spacer">
29 <div xmlns:ld="http://lambdadelta.info/" class="text">
30 <table cellpadding="4" cellspacing="0">
33 <td class="snns capitalize italic sky">
34 <a href="http://lambdadelta.info/home.html">home</a>
36 <td class="snns capitalize italic magenta">
37 <a href="http://lambdadelta.info/news.html">news</a>
39 <td class="snns capitalize italic white">
40 <a href="http://lambdadelta.info/specification.html">specification</a>
42 <td class="snnn capitalize italic white">
45 <td class="snnn capitalize italic white">
48 <td class="snns capitalize italic orange">
49 <a href="http://lambdadelta.info/documentation.html">documentation</a>
51 <td class="snns capitalize italic green">
52 <a href="http://lambdadelta.info/implementation.html">implementation</a>
54 <td class="ssnn capitalize italic green">
59 <td class="snns capitalize sky">
60 <a href="http://lambdadelta.info/home.html#foreword">foreword</a>
62 <td class="snns capitalize magenta">
63 <a href="http://lambdadelta.info/news.html#milestones">milestones</a>
65 <td class="snns capitalize white">
66 <a href="http://lambdadelta.info/specification.html#v2">version 2</a>
68 <td class="snnn capitalize white">(<a href="http://lambdadelta.info/ground_2.html">background</a> - <a href="http://lambdadelta.info/basic_2.html">core</a> - <a href="http://lambdadelta.info/apps_2.html">applications</a>)</td>
69 <td class="snnn capitalize white">
72 <td class="snns capitalize orange">
73 <a href="http://lambdadelta.info/documentation.html#v2">version 2</a>
75 <td class="snns capitalize green">
76 <a href="http://lambdadelta.info/implementation.html#helena">helena</a>
78 <td class="ssnn capitalize green">
79 <a href="http://lambdadelta.info/osn/">Open Symbolic Notation (OSN)</a>
83 <td class="snss capitalize sky">
84 <a href="http://lambdadelta.info/home.html#citations">citations</a>
86 <td class="snss capitalize magenta">
87 <a href="http://lambdadelta.info/news.html#visibility">visibility</a>
89 <td class="snss capitalize white">
90 <a href="http://lambdadelta.info/specification.html#v1">version 1</a>
92 <td class="snsn capitalize white">(<a href="http://lambdadelta.info/ground_1.html">background</a> - <a href="http://lambdadelta.info/basic_1.html">core</a>)</td>
93 <td class="snsn capitalize white">(<a href="http://lambdadelta.info/static/matita/lambdadelta/">static HELM directory</a>)</td>
94 <td class="snss capitalize orange">
95 <a href="http://lambdadelta.info/documentation.html#v1">version 1</a>
97 <td class="snss capitalize green">
98 <a href="http://lambdadelta.info/implementation.html#lddl">library</a>
100 <td class="sssn capitalize green">(<a href="http://lambdadelta.info/static/lddl/">static LDDL directory</a>)</td>
106 <section>System's Syntax and Behavior</section>
107 <body>This is a summary of the "block structure"
108 of the System's syntactic items and reductions.
110 <table name="basic_2_blk"/>
111 <body>* In terms only.
112 ** In terms and local environments only.
113 *** In global environments only.
114 **** Sort level k in terms only.
117 <div xmlns:ld="http://lambdadelta.info/" class="head2dx" id="summary">Summary of the Specification <img class="icon37" alt="[butterfly]" title="\lambda\delta butterfly" src="http://lambdadelta.info/images/b4.png" />
119 <div xmlns:ld="http://lambdadelta.info/" class="text">Here is a numerical account of the specification's contents
122 <div xmlns:ld="http://lambdadelta.info/" class="text">
123 <table cellpadding="4" cellspacing="0">
126 <td class="snns capitalize italic gray">category</td>
127 <td class="snns italic gray">objects</td>
128 <td class="snnn right italic gray">
131 <td class="snnn italic gray">
134 <td class="snnn right italic gray">
137 <td class="snnn italic gray">
140 <td class="ssnn right italic gray">
145 <td class="snns capitalize italic cyan">sizes</td>
146 <td class="snns italic cyan">files</td>
147 <td class="snnn right italic cyan">177</td>
148 <td class="snns italic cyan">characters</td>
149 <td class="snnn right italic cyan">181483</td>
150 <td class="snns italic cyan">nodes</td>
151 <td class="ssnn right italic cyan">952913</td>
154 <td class="snns capitalize italic green">propositions</td>
155 <td class="snns italic green">theorems</td>
156 <td class="snnn right italic green">49</td>
157 <td class="snns italic green">lemmas</td>
158 <td class="snnn right italic green">633</td>
159 <td class="snns italic green">total</td>
160 <td class="ssnn right italic green">682</td>
163 <td class="snss capitalize italic yellow">concepts</td>
164 <td class="snss italic yellow">declared</td>
165 <td class="snsn right italic yellow">24</td>
166 <td class="snss italic yellow">defined</td>
167 <td class="snsn right italic yellow">42</td>
168 <td class="snss italic yellow">total</td>
169 <td class="sssn right italic yellow">66</td>
174 <div xmlns:ld="http://lambdadelta.info/" class="head3sn" id="B">Stage "B"</div>
175 <ul xmlns:ld="http://lambdadelta.info/">
177 <span class="emph alpha">Ongoing.</span>
178 Context-sensitive subject equivalence
179 for native type assignment.
182 <div xmlns:ld="http://lambdadelta.info/" class="head3sn" id="A2">Stage "A2": "Extending the Applicability Condition"</div>
183 <ul xmlns:ld="http://lambdadelta.info/">
185 <span class="emph alpha">2017 January 17.</span>
186 Confluence for parallel r-transition on referred entries of local environments.
189 <ul xmlns:ld="http://lambdadelta.info/">
191 <span class="emph alpha">2016 September 15.</span>
192 Confluence for context-sensitive parallel r-transition on terms.
195 <ul xmlns:ld="http://lambdadelta.info/">
197 <span class="emph alpha">2016 April 16.</span>
198 Grammatical component reconstructed:
199 grammar, relocation, s_transition, s_computation, static
200 (anniversary milestone).
203 <ul xmlns:ld="http://lambdadelta.info/">
205 <span class="emph alpha">2016 March 25.</span>
206 Relocation with reference transforming maps (rtmap).
209 <ul xmlns:ld="http://lambdadelta.info/">
211 <span class="emph alpha">2015 October 9.</span>
212 λδ version 2A2 is started.
215 <div xmlns:ld="http://lambdadelta.info/" class="head3sn" id="A1">Stage "A1": "Extending the Applicability Condition"</div>
216 <ul xmlns:ld="http://lambdadelta.info/">
218 <span class="emph delta">2015 August 27.</span>
219 λδ version 2A1 appears too complex and is dismissed.
222 <ul xmlns:ld="http://lambdadelta.info/">
224 <span class="emph gamma">2014 October 28.</span>
225 λδ version 2A1 is released.
228 <ul xmlns:ld="http://lambdadelta.info/">
230 <span class="emph beta">2014 September 9.</span>
231 Iterated static type assignment defined (more elegantly)
232 as a primitive notion.
235 <ul xmlns:ld="http://lambdadelta.info/">
237 <span class="emph beta">2014 June 18.</span>
238 Preservation of stratified native validity
239 for context-sensitive computation on terms.
242 <ul xmlns:ld="http://lambdadelta.info/">
244 <span class="emph alpha">2014 June 9.</span>
245 Strong qrst-normalization
246 for simply typed terms.
249 <ul xmlns:ld="http://lambdadelta.info/">
251 <span class="emph alpha">2014 April 16.</span>
252 Lazy equivalence on local environments
253 added as q-step to rst-computation on closures
254 (anniversary milestone).
257 <ul xmlns:ld="http://lambdadelta.info/">
259 <span class="emph alpha">2014 January 20.</span>
260 Parametrized slicing of local environments
261 comprises both versions of this operation
262 (one from basic_1, the other used in basic_2 till now).
265 <ul xmlns:ld="http://lambdadelta.info/">
267 <span class="emph alpha">2013 August 7.</span>
268 Passive support for global environments.
271 <ul xmlns:ld="http://lambdadelta.info/">
273 <span class="emph alpha">2013 July 27.</span>
274 Reaxiomatized β-reductum as in rt-reduction.
277 <ul xmlns:ld="http://lambdadelta.info/">
279 <span class="emph alpha">2013 July 20.</span>
280 Context-sensitive strong rt-normalization
281 for simply typed terms.
284 <ul xmlns:ld="http://lambdadelta.info/">
286 <span class="emph alpha">2013 April 16.</span>
287 Reaxiomatized substitution and reduction
288 commute with respect to subclosure
289 (anniversary milestone).
292 <ul xmlns:ld="http://lambdadelta.info/">
294 <span class="emph alpha">2013 March 16.</span>
295 Mutual recursive preservation of stratified native validity
296 for rst-computation on closures.
299 <ul xmlns:ld="http://lambdadelta.info/">
301 <span class="emph alpha">2012 October 16.</span>
302 Confluence for context-free parallel reduction on closures.
305 <ul xmlns:ld="http://lambdadelta.info/">
307 <span class="emph alpha">2012 July 26.</span>
308 Term binders polarized to control ζ-reduction (not released).
311 <ul xmlns:ld="http://lambdadelta.info/">
313 <span class="emph alpha">2012 April 16.</span>
314 Context-sensitive subject equivalence
315 for atomic arity assignment
316 (anniversary milestone).
319 <ul xmlns:ld="http://lambdadelta.info/">
321 <span class="emph alpha">2012 March 15.</span>
322 Context-sensitive strong normalization
323 for simply typed terms.
326 <ul xmlns:ld="http://lambdadelta.info/">
328 <span class="emph alpha">2012 January 27.</span>
329 Support for abstract candidates of reducibility.
332 <ul xmlns:ld="http://lambdadelta.info/">
334 <span class="emph alpha">2011 September 21.</span>
335 Confluence for context-sensitive parallel reduction on terms.
338 <ul xmlns:ld="http://lambdadelta.info/">
340 <span class="emph alpha">2011 September 6.</span>
341 Confluence for context-free parallel reduction on terms.
344 <ul xmlns:ld="http://lambdadelta.info/">
346 <span class="emph alpha">2011 April 17.</span>
347 λδ version 2 is started.
350 <div xmlns:ld="http://lambdadelta.info/" class="head2dx" id="structure">Logical Structure of the Specification <img class="icon37" alt="[butterfly]" title="\lambda\delta butterfly" src="http://lambdadelta.info/images/b4.png" />
352 <div xmlns:ld="http://lambdadelta.info/" class="text">This table reports the specification's components and their planes.
354 <div xmlns:ld="http://lambdadelta.info/" class="text">
355 <table cellpadding="4" cellspacing="0">
358 <td class="snns top capitalize italic gray">component</td>
359 <td class="snns top italic gray">plane</td>
360 <td class="snns top gray">files</td>
361 <td class="snnn top gray">
364 <td class="snnn top gray">
367 <td class="ssnn top gray">
372 <td class="snns top capitalize italic water">rt-transition</td>
373 <td class="snns top italic water">t-bound context-sensitive rt-transition</td>
374 <td class="snns top water">lfpr ( ⦃?,?⦄ ⊢ ➡[?,?] ? )</td>
375 <td class="snnn top water">lfpr_length lfpr_drops lfpr_fqup lfpr_frees lfpr_aaa lfpr_lfpx lfpr_lfpr</td>
376 <td class="snnn top water">
379 <td class="ssnn top water">
384 <td class="nnns top capitalize italic water">
387 <td class="nnns top italic water">
390 <td class="snns top water">cpr ( ⦃?,?⦄ ⊢ ? ➡[?] ? )</td>
391 <td class="snnn top water">cpr_drops</td>
392 <td class="snnn top water">
395 <td class="ssnn top water">
400 <td class="nnns top capitalize italic water">
403 <td class="nnns top italic water">
406 <td class="snns top water">cpm ( ⦃?,?⦄ ⊢ ? ➡[?,?] ? )</td>
407 <td class="snnn top water">cpm_simple cpm_drops cpm_lsubr cpm_cpx</td>
408 <td class="snnn top water">
411 <td class="ssnn top water">
416 <td class="nnns top capitalize italic water">
419 <td class="snns top italic water">uncounted context-sensitive rt-transition</td>
420 <td class="snns top water">lfpx ( ⦃?,?⦄ ⊢ ⬈[?,?] ? )</td>
421 <td class="snnn top water">lfpx_length lfpx_drops lfpx_fqup lfpx_frees lfpx_aaa</td>
422 <td class="snnn top water">
425 <td class="ssnn top water">
430 <td class="nnns top capitalize italic water">
433 <td class="nnns top italic water">
436 <td class="snns top water">cpx ( ⦃?,?⦄ ⊢ ? ⬈[?] ? )</td>
437 <td class="snnn top water">cpx_simple cpx_drops cpx_lsubr</td>
438 <td class="snnn top water">
441 <td class="ssnn top water">
446 <td class="nnns top capitalize italic water">
449 <td class="snns top italic water">counted context-sensitive rt-transition</td>
450 <td class="snns top water">cpg ( ⦃?,?⦄ ⊢ ? ⬈[?,?] ? )</td>
451 <td class="snnn top water">cpg_simple cpg_drops cpg_lsubr</td>
452 <td class="snnn top water">
455 <td class="ssnn top water">
460 <td class="snns top capitalize italic green">static typing</td>
461 <td class="snns top italic green">parameters</td>
462 <td class="snns top green">sh</td>
463 <td class="snnn top green">sd</td>
464 <td class="snnn top green">
467 <td class="ssnn top green">
472 <td class="nnns top capitalize italic green">
475 <td class="snns top italic green">restricted ref. for atomic arity assignment</td>
476 <td class="snns top green">lsuba ( ? ⊢ ? ⫃⁝ ? )</td>
477 <td class="snnn top green">lsuba_drops lsuba_lsubr lsuba_aaa lsuba_lsuba</td>
478 <td class="snnn top green">
481 <td class="ssnn top green">
486 <td class="nnns top capitalize italic green">
489 <td class="snns top italic green">atomic arity assignment</td>
490 <td class="snns top green">aaa ( ⦃?,?⦄ ⊢ ? ⁝ ? )</td>
491 <td class="snnn top green">aaa_drops aaa_fqus aaa_lfeq aaa_aaa</td>
492 <td class="snnn top green">
495 <td class="ssnn top green">
500 <td class="nnns top capitalize italic green">
503 <td class="snns top italic green">equivalence for closures on referred entries</td>
504 <td class="snns top green">ffeq ( ⦃?,?,?⦄ ≡ ⦃?,?,?⦄ )</td>
505 <td class="snnn top green">ffeq_freq</td>
506 <td class="snnn top green">
509 <td class="ssnn top green">
514 <td class="nnns top capitalize italic green">
517 <td class="snns top italic green">equivalence for local environments on referred entries</td>
518 <td class="snns top green">lfeq ( ? ≡[?] ? )</td>
519 <td class="snnn top green">lfeq_length lfeq_lreq lfeq_fqup lfeq_lfeq</td>
520 <td class="snnn top green">
523 <td class="ssnn top green">
528 <td class="nnns top capitalize italic green">
531 <td class="snns top italic green">generic extension on referred entries</td>
532 <td class="snns top green">lfxs ( ? ⦻*[?,?] ? )</td>
533 <td class="snnn top green">lfxs_length lfxs_drops lfxs_fqup lfxs_lfxs</td>
534 <td class="snnn top green">
537 <td class="ssnn top green">
542 <td class="nnns top capitalize italic green">
545 <td class="snns top italic green">restricted ref. for context-sensitive free variables</td>
546 <td class="snns top green">lsubf ( ⦃?,?⦄ ⫃𝐅* ⦃?,?⦄ )</td>
547 <td class="snnn top green">lsubf_frees</td>
548 <td class="snnn top green">
551 <td class="ssnn top green">
556 <td class="nnns top capitalize italic green">
559 <td class="snns top italic green">context-sensitive free variables</td>
560 <td class="snns top green">frees ( ? ⊢ 𝐅*⦃?⦄ ≡ ? )</td>
561 <td class="snnn top green">frees_weight frees_lreq frees_drops frees_fqup frees_frees</td>
562 <td class="snnn top green">
565 <td class="ssnn top green">
570 <td class="nnns top capitalize italic green">
573 <td class="snns top italic green">restricted ref. for local env.</td>
574 <td class="snns top green">lsubr ( ? ⫃ ? )</td>
575 <td class="snnn top green">lsubr_length lsubr_drops lsubr_lsubr</td>
576 <td class="snnn top green">
579 <td class="ssnn top green">
584 <td class="snns top capitalize italic grass">s-computation</td>
585 <td class="snns top italic grass">iterated structural successor for closures</td>
586 <td class="snns top grass">fqus ( ⦃?,?,?⦄ ⊐* ⦃?,?,?⦄ )</td>
587 <td class="snnn top grass">fqus_weight fqus_drops fqus_fqup fqus_fqus</td>
588 <td class="snnn top grass">
591 <td class="ssnn top grass">
596 <td class="nnns top capitalize italic grass">
599 <td class="nnns top italic grass">
602 <td class="snns top grass">fqup ( ⦃?,?,?⦄ ⊐+ ⦃?,?,?⦄ )</td>
603 <td class="snnn top grass">fqup_weight fqup_drops fqup_fqup</td>
604 <td class="snnn top grass">
607 <td class="ssnn top grass">
612 <td class="snns top capitalize italic yellow">s-transition</td>
613 <td class="snns top italic yellow">structural successor for closures</td>
614 <td class="snns top yellow">fquq ( ⦃?,?,?⦄ ⊐⸮ ⦃?,?,?⦄ )</td>
615 <td class="snnn top yellow">fquq_length fquq_weight</td>
616 <td class="snnn top yellow">
619 <td class="ssnn top yellow">
624 <td class="nnns top capitalize italic yellow">
627 <td class="nnns top italic yellow">
630 <td class="snns top yellow">fqu ( ⦃?,?,?⦄ ⊐ ⦃?,?,?⦄ )</td>
631 <td class="snnn top yellow">fqu_length fqu_weight</td>
632 <td class="snnn top yellow">
635 <td class="ssnn top yellow">
640 <td class="snns top capitalize italic orange">relocation</td>
641 <td class="snns top italic orange">generic slicing for local environments</td>
642 <td class="snns top orange">drops_vector ( ⬇*[?,?] ? ≡ ? ) ( ⬇*[?] ? ≡ ? )</td>
643 <td class="snnn top orange">
646 <td class="snnn top orange">
649 <td class="ssnn top orange">
654 <td class="nnns top capitalize italic orange">
657 <td class="nnns top italic orange">
660 <td class="snns top orange">drops ( ⬇*[?,?] ? ≡ ? ) ( ⬇*[?] ? ≡ ? )</td>
661 <td class="snnn top orange">drops_lstar drops_weight drops_length drops_ceq drops_lexs drops_lreq drops_drops</td>
662 <td class="snnn top orange">
665 <td class="ssnn top orange">
670 <td class="nnns top capitalize italic orange">
673 <td class="snns top italic orange">generic relocation for terms</td>
674 <td class="snns top orange">lifts_vector ( ⬆*[?] ? ≡ ? )</td>
675 <td class="snnn top orange">lifts_lifts_vector</td>
676 <td class="snnn top orange">
679 <td class="ssnn top orange">
684 <td class="nnns top capitalize italic orange">
687 <td class="nnns top italic orange">
690 <td class="snns top orange">lifts ( ⬆*[?] ? ≡ ? )</td>
691 <td class="snnn top orange">lifts_simple lifts_weight lifts_lifts</td>
692 <td class="snnn top orange">
695 <td class="ssnn top orange">
700 <td class="nnns top capitalize italic orange">
703 <td class="snns top italic orange">ranged equivalence for local environments</td>
704 <td class="snns top orange">lreq ( ? ≡[?] ? )</td>
705 <td class="snnn top orange">lreq_length lreq_lreq</td>
706 <td class="snnn top orange">
709 <td class="ssnn top orange">
714 <td class="nnns top capitalize italic orange">
717 <td class="snns top italic orange">generic entrywise extension</td>
718 <td class="snns top orange">lexs ( ? ⦻*[?,?,?] ? )</td>
719 <td class="snnn top orange">lexs_length lexs_lexs</td>
720 <td class="snnn top orange">
723 <td class="ssnn top orange">
728 <td class="snns top capitalize italic red">grammar</td>
729 <td class="snns top italic red">append for local environments</td>
730 <td class="snns top red">append ( ? @@ ? )</td>
731 <td class="snnn top red">append_length</td>
732 <td class="snnn top red">
735 <td class="ssnn top red">
740 <td class="nnns top capitalize italic red">
743 <td class="snns top italic red">context-sensitive equivalences for terms</td>
744 <td class="snns top red">ceq</td>
745 <td class="snnn top red">ceq_ceq</td>
746 <td class="snnn top red">
749 <td class="ssnn top red">
754 <td class="nnns top capitalize italic red">
757 <td class="snns top italic red">same top term structure</td>
758 <td class="snns top red">tsts ( ? ≂ ? )</td>
759 <td class="snnn top red">tsts_tsts tsts_vector</td>
760 <td class="snnn top red">
763 <td class="ssnn top red">
768 <td class="nnns top capitalize italic red">
771 <td class="snns top italic red">closures</td>
772 <td class="snns top red">cl_weight ( ♯{?,?,?} )</td>
773 <td class="snnn top red">cl_restricted_weight ( ♯{?,?} )</td>
774 <td class="snnn top red">
777 <td class="ssnn top red">
782 <td class="nnns top capitalize italic red">
785 <td class="snns top italic red">internal syntax</td>
786 <td class="snns top red">genv</td>
787 <td class="snnn top red">
790 <td class="snnn top red">
793 <td class="ssnn top red">
798 <td class="nnns top capitalize italic red">
801 <td class="nnns top italic red">
804 <td class="snns top red">lenv</td>
805 <td class="snnn top red">lenv_weight ( ♯{?} )</td>
806 <td class="snnn top red">lenv_length ( |?| )</td>
807 <td class="ssnn top red">
812 <td class="nnns top capitalize italic red">
815 <td class="nnns top italic red">
818 <td class="snns top red">term</td>
819 <td class="snnn top red">term_weight ( ♯{?} )</td>
820 <td class="snnn top red">term_simple ( 𝐒⦃?⦄ )</td>
821 <td class="ssnn top red">term_vector ( Ⓐ?.? )</td>
824 <td class="nnns top capitalize italic red">
827 <td class="nnns top italic red">
830 <td class="snns top red">item</td>
831 <td class="snnn top red">
834 <td class="snnn top red">
837 <td class="ssnn top red">
842 <td class="nnss top capitalize italic red">
845 <td class="snss top italic red">external syntax</td>
846 <td class="snss top red">aarity</td>
847 <td class="snsn top red">
850 <td class="snsn top red">
853 <td class="sssn top red">
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