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<LONGLEMMAINFO name="valid-stack-leftneighbor-succ-gen"><LEMMAINFO state="proved"><LEMMANAME>valid-stack-leftneighbor-succ-gen</LEMMANAME><LEMMAGOAL><SEQGOAL><GOALSEQ><SEQ>n &lt; &#35; x, xpos = (stack = [] ⊃ 0;stack.head), xpos ≤ n
⊦ valid-stack(stack, x) ∧ leftneighbor(x, n, a) ≤ xpos → first-lower(stack, x, a) = leftneighbor(x, n, a)</SEQ></GOALSEQ></SEQGOAL></LEMMAGOAL><LEMMATYPE><USERLEMMA/></LEMMATYPE><USEDLEMMAS><LIST><LE>leftneighbor-h-le</LE><LE>leftneighbor-monotone</LE><LE>leftneighbor</LE><LE>valid-stack-rec</LE><LE>leftneighbor-zero</LE><LE>leftneighbor-h-zero</LE><LE>valid-stack-one</LE><LE>valid-stack-pop</LE><LE>leftneighbor-rec</LE><LE>leftneighbor-smaller</LE><LE>one-based-lookup</LE><LE>first-lower-rec</LE><LE>first-lower-base</LE></LIST></USEDLEMMAS><SMTINFOS><LIST></LIST></SMTINFOS><MAINGOALS><LIST></LIST></MAINGOALS><USERACTIONS>13</USERACTIONS><PROOFSTEPS>33</PROOFSTEPS><PROVEDP><T/></PROVEDP><PROOFEXISTSP><T/></PROOFEXISTSP><PROOFFILENAME>valid-stack-leftneighbor-succ-gen-proof</PROOFFILENAME><LEMMAPROOFBAG><NONE/></LEMMAPROOFBAG><SAVETREEP><F/></SAVETREEP><INFOFILENAME>valid-stack-leftneighbor-succ-gen-proof-info</INFOFILENAME><LEMMAPROOFINFOBAG><NONE/></LEMMAPROOFINFOBAG><SAVEINFOSP><F/></SAVEINFOSP><SIMPFEATURES><LIST></LIST></SIMPFEATURES><HISTINFO><LIST></LIST></HISTINFO></LEMMAINFO><VALIDITY>The proof is valid.</VALIDITY><FEATURES><LIST></LIST></FEATURES><HISTORY><LIST></LIST></HISTORY><USEDBY><LIST><LE>valid-stack-leftneighbor-succ</LE></LIST></USEDBY><USEDBYUNITS><LIST></LIST></USEDBYUNITS><COMMENT></COMMENT><USED_SPECLEMS></USED_SPECLEMS><USED_SIMPRULES><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/unit.xml</HTMLTARGET><HTMLSTRING>nat</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/fle-01/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>fle-01</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/list-data/export/unit.xml</HTMLTARGET><HTMLSTRING>list-data</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/list-data/export/l/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>l</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/unit.xml</HTMLTARGET><HTMLSTRING>nat</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/let/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>let</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/unit.xml</HTMLTARGET><HTMLSTRING>nat-basic</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/f/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>f</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/unit.xml</HTMLTARGET><HTMLSTRING>nat</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/lez-01/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>lez-01</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/list/export/unit.xml</HTMLTARGET><HTMLSTRING>list</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME>natlist</INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/list/export/a/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>a</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/list/export/unit.xml</HTMLTARGET><HTMLSTRING>list</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME>natlist</INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/list/export/c-08/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>c-08</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/list/export/unit.xml</HTMLTARGET><HTMLSTRING>list</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME>natlist</INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/list/export/c/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>c</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/list/export/unit.xml</HTMLTARGET><HTMLSTRING>list</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME>natlist</INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/list/export/e-01/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>e-01</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/unit.xml</HTMLTARGET><HTMLSTRING>nat</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/pred-03/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>pred-03</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/unit.xml</HTMLTARGET><HTMLSTRING>nat</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/lels-03/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>lels-03</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/unit.xml</HTMLTARGET><HTMLSTRING>nat-basic</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/succ/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>succ</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/unit.xml</HTMLTARGET><HTMLSTRING>nat</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/fle/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>fle</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/unit.xml</HTMLTARGET><HTMLSTRING>nat-basic</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/lf/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>lf</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/unit.xml</HTMLTARGET><HTMLSTRING>nat</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/lels-02/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>lels-02</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/unit.xml</HTMLTARGET><HTMLSTRING>nat</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/lez/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>lez</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/unit.xml</HTMLTARGET><HTMLSTRING>nat-basic</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/zero/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>zero</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/unit.xml</HTMLTARGET><HTMLSTRING>nat-basic</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/less-1/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>less-1</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/unit.xml</HTMLTARGET><HTMLSTRING>nat</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/elim-pred-c/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>elim-pred-c</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/oelem/export/unit.xml</HTMLTARGET><HTMLSTRING>oelem</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME>natlist</INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/oelem/export/transitivity/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>transitivity</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/unit.xml</HTMLTARGET><HTMLSTRING>nat</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/lels/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>lels</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/unit.xml</HTMLTARGET><HTMLSTRING>nat-basic</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/c/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>c</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/unit.xml</HTMLTARGET><HTMLSTRING>nat-basic</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/a/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>a</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/unit.xml</HTMLTARGET><HTMLSTRING>nat-basic</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/s-04/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>s-04</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/unit.xml</HTMLTARGET><HTMLSTRING>nat</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/succ-01/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>succ-01</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/unit.xml</HTMLTARGET><HTMLSTRING>nat-basic</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat-basic/export/oz/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>oz</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/unit.xml</HTMLTARGET><HTMLSTRING>nat</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/le-ls-simp/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>le-ls-simp</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/unit.xml</HTMLTARGET><HTMLSTRING>nat</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/ls-le-simp/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>ls-le-simp</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/list/export/unit.xml</HTMLTARGET><HTMLSTRING>list</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME>natlist</INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/list/export/app/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>app</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/unit.xml</HTMLTARGET><HTMLSTRING>nat</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/le-one/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>le-one</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/oelem/export/unit.xml</HTMLTARGET><HTMLSTRING>oelem</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/oelem/export/nll/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>nll</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/oelem/export/unit.xml</HTMLTARGET><HTMLSTRING>oelem</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/oelem/export/nnn/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>nnn</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/unit.xml</HTMLTARGET><HTMLSTRING>nat</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME></INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/nat/export/le-def/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>le-def</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ><USED_SEQ><SPECNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/oelem/export/unit.xml</HTMLTARGET><HTMLSTRING>oelem</HTMLSTRING></HTMLLINK></SPECNAME><INSTNAME>natlist</INSTNAME><LEMNAME><HTMLLINK><HTMLTARGET>../../../../../../lib/basic/specs/oelem/export/sls/longlemmainfo.xml</HTMLTARGET><HTMLSTRING>sls</HTMLSTRING></HTMLLINK></LEMNAME></USED_SEQ></USED_SIMPRULES></LONGLEMMAINFO>