--- a/thys2/Journal/Paper.tex Mon Nov 01 10:52:44 2021 +0000
+++ b/thys2/Journal/Paper.tex Tue Nov 02 13:57:59 2021 +0000
@@ -47,7 +47,7 @@
%
\isatagdocument
%
-\isamarkupsection{Introduction%
+\isamarkupsection{Core of the proof%
}
\isamarkuptrue%
%
@@ -69,12 +69,93 @@
In the previous work, Ausaf and Urban established the below equality:
\begin{lemma}
-\isa{{\normalsize{}If\,}\ v\ {\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r\mbox{$^\downarrow$}{\isacharparenright}{\kern0pt}{\isacharbackslash}{\kern0pt}c\ {\normalsize \,then\,}\ retrieve\ {\isacharparenleft}{\kern0pt}r\mbox{$\bbslash$}c{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ retrieve\ r\ {\isacharparenleft}{\kern0pt}inj\ {\isacharparenleft}{\kern0pt}r\mbox{$^\downarrow$}{\isacharparenright}{\kern0pt}\ c\ v{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}}
+\isa{{\normalsize{}If\,}\ v\ {\isacharcolon}{\kern0pt}\ r{\isacharbackslash}{\kern0pt}s\ {\normalsize \,then\,}\ Some\ {\isacharparenleft}{\kern0pt}flex\ r\ id\ s\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode\ {\isacharparenleft}{\kern0pt}retrieve\ {\isacharparenleft}{\kern0pt}r\mbox{$^\uparrow$}\mbox{$\bbslash$}s{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ r{\isachardot}{\kern0pt}}
+\end{lemma}
+
+This lemma establishes a link with the lexer without bit-codes.
+
+With it we get the correctness of bit-coded algorithm.
+\begin{lemma}
+\isa{lexer\mbox{$_b$}\ r\ s\ {\isacharequal}{\kern0pt}\ lexer\ r\ s}
\end{lemma}
-This lemma links the derivative of a bit-coded regular expression with
-the regular expression itself before the derivative.
+However what is not certain is whether we can add simplification
+to the bit-coded algorithm, without breaking the correct lexing output.
+This might sound trivial in the case of producing a YES/NO answer,
+but once we require a lexing output to be produced (which is required
+in applications like compiler front-end, malicious attack domain extraction,
+etc.), it is not straightforward if we still extract what is needed according
+to the POSIX standard.
+
+By simplification, we mean specifically the following rules:
+
+\begin{center}
+ \begin{tabular}{lcl}
+ \isa{\mbox{}\inferrule{\mbox{}}{\mbox{ASEQ\ bs\ AZERO\ r\isactrlsub {\isadigit{2}}\ {\isasymleadsto}\ AZERO}}}\\
+ \isa{\mbox{}\inferrule{\mbox{}}{\mbox{ASEQ\ bs\ r\isactrlsub {\isadigit{1}}\ AZERO\ {\isasymleadsto}\ AZERO}}}\\
+ \isa{\mbox{}\inferrule{\mbox{}}{\mbox{ASEQ\ bs\ {\isacharparenleft}{\kern0pt}AONE\ bs\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isasymleadsto}\ fuse\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ bs\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ r\isactrlsub {\isadigit{1}}}}}\\
+ \isa{\mbox{}\inferrule{\mbox{bs\ {\isasymleadsto}\ r\isactrlsub {\isadigit{1}}}}{\mbox{ASEQ\ bs\ bs\ r{\isadigit{3}}{\isachardot}{\kern0pt}{\isadigit{0}}\ {\isasymleadsto}\ ASEQ\ bs\ r\isactrlsub {\isadigit{1}}\ r{\isadigit{3}}{\isachardot}{\kern0pt}{\isadigit{0}}}}}\\
+ \isa{\mbox{}\inferrule{\mbox{bs\ {\isasymleadsto}\ r\isactrlsub {\isadigit{2}}}}{\mbox{ASEQ\ bs\ r{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ bs\ {\isasymleadsto}\ ASEQ\ bs\ r{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ r\isactrlsub {\isadigit{2}}}}}\\
+ \isa{\mbox{}\inferrule{\mbox{bs\ {\isasymleadsto}\ r\isactrlsub {\isadigit{1}}}}{\mbox{AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}bs{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs{\isadigit{2}}{\isachardot}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs{\isadigit{2}}{\isachardot}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}}}}\\
+ \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}\ {\isacharat}{\kern0pt}\ AZERO\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}rsb{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}\ {\isacharat}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}}}}\\
+ \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ AALTs\ bs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ rs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}rsb{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ rs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ {\isacharat}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}}}}\\
+ \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fuse\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ rs}}}\\
+ \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fuse\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}}}}\\
+ \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymleadsto}\ AZERO}}}\\
+ \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharbrackleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isasymleadsto}\ fuse\ bs\ r\isactrlsub {\isadigit{1}}}}}\\
+ \isa{\mbox{}\inferrule{\mbox{bs\mbox{$^\downarrow$}\ {\isacharequal}{\kern0pt}\ r\isactrlsub {\isadigit{2}}\mbox{$^\downarrow$}}}{\mbox{AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}bs{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rsb\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rsc{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}bs{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rsb\ {\isacharat}{\kern0pt}\ rsc{\isacharparenright}{\kern0pt}}}}\\
+
+
+ \isa{\mbox{}\inferrule{\mbox{}}{\mbox{Empty\ {\isacharcolon}{\kern0pt}\ \isactrlbold {\isadigit{1}}}}} &
+ \isa{\mbox{}\inferrule{\mbox{}}{\mbox{Char\ c\ {\isacharcolon}{\kern0pt}\ c}}}\\[4mm]
+ \isa{\mbox{}\inferrule{\mbox{v\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{1}}}}{\mbox{Left\ v\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}}}} &
+ \isa{\mbox{}\inferrule{\mbox{v\isactrlsub {\isadigit{2}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{1}}}}{\mbox{Right\ v\isactrlsub {\isadigit{2}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{2}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{1}}}}}\\[4mm]
+ \isa{\mbox{}\inferrule{\mbox{v\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{1}}}\\\ \mbox{v\isactrlsub {\isadigit{2}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{2}}}}{\mbox{Seq\ v\isactrlsub {\isadigit{1}}\ v\isactrlsub {\isadigit{2}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}}}} &
+ \isa{\mbox{}\inferrule{\mbox{{\isasymforall}v{\isasymin}vs{\isachardot}{\kern0pt}\ v\ {\isacharcolon}{\kern0pt}\ r\ {\isasymand}\ {\isacharbar}{\kern0pt}v{\isacharbar}{\kern0pt}\ {\isasymnoteq}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}}}{\mbox{Stars\ vs\ {\isacharcolon}{\kern0pt}\ r\isactrlsup {\isasymstar}}}}
+
+
+ \end{tabular}
+\end{center}
+
+And these can be made compact by the following simplification function:
+
+\begin{center}
+ \begin{tabular}{lcl}
+ \isa{bsimp\ {\isacharparenleft}{\kern0pt}ASEQ\ bs\ r\isactrlsub {\isadigit{1}}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{bsimp{\isacharunderscore}{\kern0pt}ASEQ\ bs\ {\isacharparenleft}{\kern0pt}bsimp\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bsimp\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}\\
+ \isa{bsimp\ {\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ rs{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{bsimp{\isacharunderscore}{\kern0pt}AALTs\ bs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ {\isacharparenleft}{\kern0pt}distinctBy\ {\isacharparenleft}{\kern0pt}flts\ {\isacharparenleft}{\kern0pt}map\ bsimp\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ erase\ {\isasymemptyset}{\isacharparenright}{\kern0pt}}\\
+ \isa{bsimp\ AZERO} & $\dn$ & \isa{AZERO}\\
+ \isa{{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\mbox{$^\uparrow$}} & $\dn$ & \isa{AALT\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fuse\ {\isacharbrackleft}{\kern0pt}Z{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\mbox{$^\uparrow$}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fuse\ {\isacharbrackleft}{\kern0pt}S{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}\mbox{$^\uparrow$}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}}\\
+\end{tabular}
+\end{center}
+
+The core idea of the proof is that two regular expressions,
+if "isomorphic" up to a finite number of rewrite steps, will
+remain so when we take derivative on both of them.
+This can be expressed by the following rewrite relation lemma:
+\begin{lemma}
+\isa{{\isacharparenleft}{\kern0pt}r\mbox{$\bbslash$}s{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ bders{\isacharunderscore}{\kern0pt}simp\ r\ s}
+\end{lemma}%
+\end{isamarkuptext}\isamarkuptrue%
+%
+\isadelimdocument
+%
+\endisadelimdocument
+%
+\isatagdocument
+%
+\isamarkupsection{Introduction%
+}
+\isamarkuptrue%
+%
+\endisatagdocument
+{\isafolddocument}%
+%
+\isadelimdocument
+%
+\endisadelimdocument
+%
+\begin{isamarkuptext}%
Brzozowski \cite{Brzozowski1964} introduced the notion of the {\em
derivative} \isa{r{\isacharbackslash}{\kern0pt}c} of a regular expression \isa{r} w.r.t.\
a character~\isa{c}, and showed that it gave a simple solution to the
@@ -1586,1311 +1667,1311 @@
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\endinput
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-%:%1491=1860%:%
-%:%1492=1861%:%
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-%:%1555=1912%:%
-%:%1556=1913%:%
-%:%1557=1914%:%
-%:%1558=1915%:%
-%:%1559=1916%:%
+%:%1410=1489%:%
+%:%1422=1493%:%
+%:%1423=1494%:%
+%:%1424=1495%:%
+%:%1425=1496%:%
+%:%1426=1497%:%
+%:%1427=1498%:%
+%:%1428=1499%:%
+%:%1429=1500%:%
+%:%1430=1501%:%
+%:%1431=1502%:%
+%:%1432=1503%:%
+%:%1433=1504%:%
+%:%1434=1505%:%
+%:%1435=1506%:%
+%:%1436=1507%:%
+%:%1437=1508%:%
+%:%1438=1509%:%
+%:%1439=1510%:%
+%:%1440=1511%:%
+%:%1441=1512%:%
+%:%1442=1513%:%
+%:%1443=1514%:%
+%:%1444=1515%:%
+%:%1445=1516%:%
+%:%1446=1517%:%
+%:%1447=1518%:%
+%:%1448=1519%:%
+%:%1449=1520%:%
+%:%1450=1521%:%
+%:%1451=1522%:%
+%:%1452=1523%:%
+%:%1453=1524%:%
+%:%1454=1525%:%
+%:%1455=1526%:%
+%:%1456=1527%:%
+%:%1457=1528%:%
+%:%1458=1529%:%
+%:%1459=1530%:%
+%:%1460=1531%:%
+%:%1461=1532%:%
+%:%1462=1533%:%
+%:%1463=1534%:%
+%:%1464=1535%:%
+%:%1465=1536%:%
+%:%1466=1537%:%
+%:%1467=1538%:%
+%:%1468=1539%:%
+%:%1469=1540%:%
+%:%1470=1541%:%
+%:%1471=1542%:%
+%:%1472=1543%:%
+%:%1473=1544%:%
+%:%1474=1545%:%
+%:%1475=1546%:%
+%:%1476=1547%:%
+%:%1477=1548%:%
+%:%1478=1549%:%
+%:%1479=1550%:%
+%:%1480=1551%:%
+%:%1481=1552%:%
+%:%1482=1553%:%
+%:%1483=1554%:%
+%:%1484=1555%:%
+%:%1485=1556%:%
+%:%1486=1557%:%
+%:%1487=1558%:%
+%:%1488=1559%:%
+%:%1489=1560%:%
+%:%1490=1561%:%
+%:%1491=1562%:%
+%:%1492=1563%:%
+%:%1493=1564%:%
+%:%1494=1565%:%
+%:%1495=1566%:%
+%:%1496=1567%:%
+%:%1497=1568%:%
+%:%1498=1569%:%
+%:%1499=1570%:%
+%:%1500=1571%:%
+%:%1501=1572%:%
+%:%1502=1573%:%
+%:%1503=1574%:%
+%:%1504=1575%:%
+%:%1505=1576%:%
+%:%1506=1577%:%
+%:%1507=1578%:%
+%:%1508=1579%:%
+%:%1509=1580%:%
+%:%1510=1581%:%
+%:%1511=1582%:%
+%:%1511=1698%:%
+%:%1512=1699%:%
+%:%1513=1700%:%
+%:%1514=1701%:%
+%:%1515=1702%:%
+%:%1515=1863%:%
+%:%1516=1864%:%
+%:%1517=1865%:%
+%:%1518=1866%:%
+%:%1519=1867%:%
+%:%1520=1868%:%
+%:%1521=1869%:%
+%:%1522=1870%:%
+%:%1523=1871%:%
+%:%1524=1872%:%
+%:%1525=1873%:%
+%:%1526=1874%:%
+%:%1527=1875%:%
+%:%1528=1876%:%
+%:%1529=1877%:%
+%:%1530=1878%:%
+%:%1531=1879%:%
+%:%1532=1880%:%
+%:%1533=1881%:%
+%:%1534=1882%:%
+%:%1534=1883%:%
+%:%1535=1884%:%
+%:%1535=1885%:%
+%:%1535=1886%:%
+%:%1536=1887%:%
+%:%1536=1888%:%
+%:%1537=1889%:%
+%:%1538=1890%:%
+%:%1539=1891%:%
+%:%1540=1892%:%
+%:%1541=1893%:%
+%:%1542=1894%:%
+%:%1543=1895%:%
+%:%1544=1896%:%
+%:%1545=1897%:%
+%:%1546=1898%:%
+%:%1547=1899%:%
+%:%1548=1900%:%
+%:%1549=1901%:%
+%:%1550=1902%:%
+%:%1551=1903%:%
+%:%1552=1904%:%
+%:%1553=1905%:%
+%:%1553=1906%:%
+%:%1554=1907%:%
+%:%1555=1908%:%
+%:%1556=1909%:%
+%:%1556=1910%:%
+%:%1556=1911%:%
+%:%1557=1912%:%
+%:%1558=1913%:%
+%:%1559=1914%:%
+%:%1559=1915%:%
+%:%1560=1916%:%
%:%1560=1917%:%
%:%1561=1918%:%
%:%1562=1919%:%
@@ -3058,4 +3139,66 @@
%:%1567=1924%:%
%:%1568=1925%:%
%:%1569=1926%:%
-%:%1582=1932%:%
\ No newline at end of file
+%:%1570=1927%:%
+%:%1571=1928%:%
+%:%1572=1929%:%
+%:%1573=1930%:%
+%:%1574=1931%:%
+%:%1575=1932%:%
+%:%1576=1933%:%
+%:%1577=1934%:%
+%:%1586=1939%:%
+%:%1598=1943%:%
+%:%1599=1944%:%
+%:%1600=1945%:%
+%:%1601=1946%:%
+%:%1602=1947%:%
+%:%1603=1948%:%
+%:%1604=1949%:%
+%:%1605=1950%:%
+%:%1606=1951%:%
+%:%1607=1952%:%
+%:%1608=1953%:%
+%:%1609=1954%:%
+%:%1610=1955%:%
+%:%1611=1956%:%
+%:%1612=1957%:%
+%:%1613=1958%:%
+%:%1614=1959%:%
+%:%1615=1960%:%
+%:%1616=1961%:%
+%:%1617=1962%:%
+%:%1618=1963%:%
+%:%1619=1964%:%
+%:%1620=1965%:%
+%:%1621=1966%:%
+%:%1622=1967%:%
+%:%1623=1968%:%
+%:%1624=1969%:%
+%:%1625=1970%:%
+%:%1626=1971%:%
+%:%1627=1972%:%
+%:%1628=1973%:%
+%:%1629=1974%:%
+%:%1630=1975%:%
+%:%1631=1976%:%
+%:%1632=1977%:%
+%:%1633=1978%:%
+%:%1634=1979%:%
+%:%1635=1980%:%
+%:%1636=1981%:%
+%:%1637=1982%:%
+%:%1638=1983%:%
+%:%1639=1984%:%
+%:%1640=1985%:%
+%:%1641=1986%:%
+%:%1642=1987%:%
+%:%1643=1988%:%
+%:%1644=1989%:%
+%:%1645=1990%:%
+%:%1646=1991%:%
+%:%1647=1992%:%
+%:%1648=1993%:%
+%:%1649=1994%:%
+%:%1650=1995%:%
+%:%1663=2001%:%
\ No newline at end of file
--- a/thys2/Journal/Paper.thy Mon Nov 01 10:52:44 2021 +0000
+++ b/thys2/Journal/Paper.thy Tue Nov 02 13:57:59 2021 +0000
@@ -131,16 +131,11 @@
(*>*)
-
-
-section \<open>Introduction\<close>
-
-
+section\<open>Core of the proof\<close>
text \<open>
-
-This works builds on previous work by Ausaf and Urban using
+This paper builds on previous work by Ausaf and Urban using
regular expression'd bit-coded derivatives to do lexing that
-is both fast and satisfied the POSIX specification.
+is both fast and satisfies the POSIX specification.
In their work, a bit-coded algorithm introduced by Sulzmann and Lu
was formally verified in Isabelle, by a very clever use of
flex function and retrieve to carefully mimic the way a value is
@@ -148,11 +143,102 @@
In the previous work, Ausaf and Urban established the below equality:
\begin{lemma}
-@{thm [mode=IfThen] bder_retrieve}
+@{thm [mode=IfThen] MAIN_decode}
+\end{lemma}
+
+This lemma establishes a link with the lexer without bit-codes.
+
+With it we get the correctness of bit-coded algorithm.
+\begin{lemma}
+@{thm [mode=IfThen] blexer_correctness}
\end{lemma}
-This lemma links the derivative of a bit-coded regular expression with
-the regular expression itself before the derivative.
+However what is not certain is whether we can add simplification
+to the bit-coded algorithm, without breaking the correct lexing output.
+This might sound trivial in the case of producing a YES/NO answer,
+but once we require a lexing output to be produced (which is required
+in applications like compiler front-end, malicious attack domain extraction,
+etc.), it is not straightforward if we still extract what is needed according
+to the POSIX standard.
+
+By simplification, we mean specifically the following rules:
+
+\begin{center}
+ \begin{tabular}{lcl}
+ @{thm[mode=Axiom] rrewrite.intros(1)[of "bs" "r\<^sub>2"]}\\
+ @{thm[mode=Axiom] rrewrite.intros(2)[of "bs" "r\<^sub>1"]}\\
+ @{thm[mode=Axiom] rrewrite.intros(3)[of "bs" "bs\<^sub>1" "r\<^sub>1"]}\\
+ @{thm[mode=Rule] rrewrite.intros(4)[of "r\<^sub>1" "r\<^sub>2" "bs" "r\<^sub>3"]}\\
+ @{thm[mode=Rule] rrewrite.intros(5)[of "bs" "r\<^sub>2"]}\\
+ @{thm[mode=Rule] rrewrite.intros(6)[of "bs" "r\<^sub>1"]}\\
+ @{thm[mode=Axiom] rrewrite.intros(7)[of "bs" "r\<^sub>2"]}\\
+ @{thm[mode=Axiom] rrewrite.intros(8)[of "bs" "r\<^sub>1"]}\\
+ @{thm[mode=Axiom] rrewrite.intros(9)[of "bs" "r\<^sub>2"]}\\
+ @{thm[mode=Axiom] rrewrite.intros(10)[of "bs" "r\<^sub>1"]}\\
+ @{thm[mode=Axiom] rrewrite.intros(11)[of "bs" ]}\\
+ @{thm[mode=Axiom] rrewrite.intros(12)[of "bs" "r\<^sub>1"]}\\
+ @{thm[mode=Rule] rrewrite.intros(13)[of "a\<^sub>1" "a\<^sub>2" "bs" "rs\<^sub>a" "rs\<^sub>b" "rs\<^sub>c"]}\\
+
+ \end{tabular}
+\end{center}
+
+
+And these can be made compact by the following simplification function:
+
+\begin{center}
+ \begin{tabular}{lcl}
+ @{thm (lhs) bsimp.simps(1)[of "bs" "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) bsimp.simps(1)[of "bs" "r\<^sub>1" "r\<^sub>2"]}\\
+ @{thm (lhs) bsimp.simps(2)} & $\dn$ & @{thm (rhs) bsimp.simps(2)}\\
+ @{thm (lhs) bsimp.simps(3)} & $\dn$ & @{thm (rhs) bsimp.simps(3)}\\
+ @{thm (lhs) intern.simps(4)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) intern.simps(4)[of "r\<^sub>1" "r\<^sub>2"]}\\
+\end{tabular}
+\end{center}
+
+where the function $\mathit{bsimp_AALTs}$
+
+The core idea of the proof is that two regular expressions,
+if "isomorphic" up to a finite number of rewrite steps, will
+remain "isomorphic" when we take the same sequence of
+derivatives on both of them.
+This can be expressed by the following rewrite relation lemma:
+\begin{lemma}
+@{thm [mode=IfThen] central}
+\end{lemma}
+
+This isomorphic relation implies a property that leads to the
+correctness result:
+if two (nullable) regular expressions are "rewritable" in many steps
+from one another,
+then a call to function $textit{bmkeps}$ gives the same
+bit-sequence :
+\begin{lemma}
+@{thm [mode=IfThen] rewrites_bmkeps}
+\end{lemma}
+
+Given the same bit-sequence, the decode function
+will give out the same value, which is the output
+of both lexers:
+\begin{lemma}
+@{thm blexer_def}
+\end{lemma}
+
+\begin{lemma}
+@{thm blexer_simp_def}
+\end{lemma}
+
+And that yields the correctness result:
+\begin{lemma}
+@{thm blexersimp_correctness}
+\end{lemma}
+
+
+\<close>
+
+section \<open>Introduction\<close>
+
+
+text \<open>
+
Brzozowski \cite{Brzozowski1964} introduced the notion of the {\em
derivative} @{term "der c r"} of a regular expression \<open>r\<close> w.r.t.\
@@ -332,6 +418,9 @@
\<close>
+
+
+
section \<open>Preliminaries\<close>
text \<open>\noindent Strings in Isabelle/HOL are lists of characters with
--- a/thys2/Journal/SizeBound.tex Mon Nov 01 10:52:44 2021 +0000
+++ b/thys2/Journal/SizeBound.tex Tue Nov 02 13:57:59 2021 +0000
@@ -1,7263 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{SizeBound}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ SizeBound\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Lexer{\isachardoublequoteclose}\ \isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Bit-Encodings%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{datatype}\isamarkupfalse%
-\ bit\ {\isacharequal}{\kern0pt}\ Z\ {\isacharbar}{\kern0pt}\ S\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}val\ {\isasymRightarrow}\ bit\ list{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}code\ Void\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}code\ {\isacharparenleft}{\kern0pt}Char\ c{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}code\ {\isacharparenleft}{\kern0pt}Left\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Z\ {\isacharhash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}code\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}code\ {\isacharparenleft}{\kern0pt}Right\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ S\ {\isacharhash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}code\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}code\ {\isacharparenleft}{\kern0pt}Seq\ v{\isadigit{1}}\ v{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}code\ v{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}\ {\isacharparenleft}{\kern0pt}code\ v{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}code\ {\isacharparenleft}{\kern0pt}Stars\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}S{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}code\ {\isacharparenleft}{\kern0pt}Stars\ {\isacharparenleft}{\kern0pt}v\ {\isacharhash}{\kern0pt}\ vs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ \ {\isacharparenleft}{\kern0pt}Z\ {\isacharhash}{\kern0pt}\ code\ v{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}\ code\ {\isacharparenleft}{\kern0pt}Stars\ vs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ \isanewline
-\ \ Stars{\isacharunderscore}{\kern0pt}add\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}val\ {\isasymRightarrow}\ val\ {\isasymRightarrow}\ val{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}Stars{\isacharunderscore}{\kern0pt}add\ v\ {\isacharparenleft}{\kern0pt}Stars\ vs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Stars\ {\isacharparenleft}{\kern0pt}v\ {\isacharhash}{\kern0pt}\ vs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{function}\isamarkupfalse%
-\isanewline
-\ \ decode{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}bit\ list\ {\isasymRightarrow}\ rexp\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}val\ {\isacharasterisk}{\kern0pt}\ bit\ list{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}\ ds\ ZERO\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Void{\isacharcomma}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}\ ds\ ONE\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Void{\isacharcomma}{\kern0pt}\ ds{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}\ ds\ {\isacharparenleft}{\kern0pt}CH\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Char\ d{\isacharcomma}{\kern0pt}\ ds{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}ALT\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Void{\isacharcomma}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Z\ {\isacharhash}{\kern0pt}\ ds{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}ALT\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ ds{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode{\isacharprime}{\kern0pt}\ ds\ r{\isadigit{1}}\ in\ {\isacharparenleft}{\kern0pt}Left\ v{\isacharcomma}{\kern0pt}\ ds{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}S\ {\isacharhash}{\kern0pt}\ ds{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}ALT\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ ds{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode{\isacharprime}{\kern0pt}\ ds\ r{\isadigit{2}}\ in\ {\isacharparenleft}{\kern0pt}Right\ v{\isacharcomma}{\kern0pt}\ ds{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}\ ds\ {\isacharparenleft}{\kern0pt}SEQ\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}v{\isadigit{1}}{\isacharcomma}{\kern0pt}\ ds{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode{\isacharprime}{\kern0pt}\ ds\ r{\isadigit{1}}\ in\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ let\ {\isacharparenleft}{\kern0pt}v{\isadigit{2}}{\isacharcomma}{\kern0pt}\ ds{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode{\isacharprime}{\kern0pt}\ ds{\isacharprime}{\kern0pt}\ r{\isadigit{2}}\ in\ {\isacharparenleft}{\kern0pt}Seq\ v{\isadigit{1}}\ v{\isadigit{2}}{\isacharcomma}{\kern0pt}\ ds{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}STAR\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Void{\isacharcomma}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}S\ {\isacharhash}{\kern0pt}\ ds{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}STAR\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Stars\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharcomma}{\kern0pt}\ ds{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Z\ {\isacharhash}{\kern0pt}\ ds{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}STAR\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ ds{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode{\isacharprime}{\kern0pt}\ ds\ r\ in\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ let\ {\isacharparenleft}{\kern0pt}vs{\isacharcomma}{\kern0pt}\ ds{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode{\isacharprime}{\kern0pt}\ ds{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}STAR\ r{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ in\ {\isacharparenleft}{\kern0pt}Stars{\isacharunderscore}{\kern0pt}add\ v\ vs{\isacharcomma}{\kern0pt}\ ds{\isacharprime}{\kern0pt}{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ pat{\isacharunderscore}{\kern0pt}completeness\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ decode{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}smaller{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}dom\ {\isacharparenleft}{\kern0pt}ds{\isacharcomma}{\kern0pt}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}length\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}decode{\isacharprime}{\kern0pt}\ ds\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymle}\ length\ ds{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ ds\ r{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ decode{\isacharprime}{\kern0pt}{\isachardot}{\kern0pt}psimps\ split{\isacharcolon}{\kern0pt}\ prod{\isachardot}{\kern0pt}split{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{using}\isamarkupfalse%
-\ dual{\isacharunderscore}{\kern0pt}order{\isachardot}{\kern0pt}trans\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ dual{\isacharunderscore}{\kern0pt}order{\isachardot}{\kern0pt}trans\ le{\isacharunderscore}{\kern0pt}SucI{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{termination}\isamarkupfalse%
-\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\ \ \isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}relation\ {\isachardoublequoteopen}inv{\isacharunderscore}{\kern0pt}image\ {\isacharparenleft}{\kern0pt}measure{\isacharparenleft}{\kern0pt}{\isacharpercent}{\kern0pt}cs{\isachardot}{\kern0pt}\ size\ cs{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}{\isacharasterisk}{\kern0pt}lex{\isacharasterisk}{\kern0pt}{\isachargreater}{\kern0pt}\ measure{\isacharparenleft}{\kern0pt}{\isacharpercent}{\kern0pt}s{\isachardot}{\kern0pt}\ size\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharpercent}{\kern0pt}{\isacharparenleft}{\kern0pt}ds{\isacharcomma}{\kern0pt}r{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharcomma}{\kern0pt}ds{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\ \isanewline
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ dest{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ decode{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}smaller{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ less{\isacharunderscore}{\kern0pt}Suc{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}le\ snd{\isacharunderscore}{\kern0pt}conv{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\isanewline
-\ \ decode\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}bit\ list\ {\isasymRightarrow}\ rexp\ {\isasymRightarrow}\ val\ option{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}decode\ ds\ r\ {\isasymequiv}\ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ ds{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode{\isacharprime}{\kern0pt}\ ds\ r\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ in\ {\isacharparenleft}{\kern0pt}if\ ds{\isacharprime}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ then\ Some\ v\ else\ None{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ decode{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}Stars{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymforall}v{\isasymin}set\ vs{\isachardot}{\kern0pt}\ {\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isachardot}{\kern0pt}\ decode{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}code\ v\ {\isacharat}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ flat\ v\ {\isasymnoteq}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}Stars\ vs{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}\ ds{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}STAR\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Stars\ vs{\isacharcomma}{\kern0pt}\ ds{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ vs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ decode{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}code{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}decode{\isacharprime}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}code\ v{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}\ ds{\isacharparenright}{\kern0pt}\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}v{\isacharcomma}{\kern0pt}\ ds{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ v\ r\ arbitrary{\isacharcolon}{\kern0pt}\ ds{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ decode{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}Stars\ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ decode{\isacharunderscore}{\kern0pt}code{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}decode\ {\isacharparenleft}{\kern0pt}code\ v{\isacharparenright}{\kern0pt}\ r\ {\isacharequal}{\kern0pt}\ Some\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ \isacommand{unfolding}\isamarkupfalse%
-\ decode{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ append{\isacharunderscore}{\kern0pt}Nil{\isadigit{2}}\ decode{\isacharprime}{\kern0pt}{\isacharunderscore}{\kern0pt}code\ old{\isachardot}{\kern0pt}prod{\isachardot}{\kern0pt}case{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Annotated Regular Expressions%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{datatype}\isamarkupfalse%
-\ arexp\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ AZERO\isanewline
-{\isacharbar}{\kern0pt}\ AONE\ {\isachardoublequoteopen}bit\ list{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ ACHAR\ {\isachardoublequoteopen}bit\ list{\isachardoublequoteclose}\ char\isanewline
-{\isacharbar}{\kern0pt}\ ASEQ\ {\isachardoublequoteopen}bit\ list{\isachardoublequoteclose}\ arexp\ arexp\isanewline
-{\isacharbar}{\kern0pt}\ AALTs\ {\isachardoublequoteopen}bit\ list{\isachardoublequoteclose}\ {\isachardoublequoteopen}arexp\ list{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ ASTAR\ {\isachardoublequoteopen}bit\ list{\isachardoublequoteclose}\ arexp\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\isanewline
-\ \ {\isachardoublequoteopen}AALT\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}\ {\isasymequiv}\ AALTs\ bs\ {\isacharbrackleft}{\kern0pt}r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ asize\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}asize\ AZERO\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}asize\ {\isacharparenleft}{\kern0pt}AONE\ cs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}asize\ {\isacharparenleft}{\kern0pt}ACHAR\ cs\ c{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}asize\ {\isacharparenleft}{\kern0pt}AALTs\ cs\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}sum{\isacharunderscore}{\kern0pt}list\ {\isacharparenleft}{\kern0pt}map\ asize\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}asize\ {\isacharparenleft}{\kern0pt}ASEQ\ cs\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}asize\ r{\isadigit{1}}\ {\isacharplus}{\kern0pt}\ asize\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}asize\ {\isacharparenleft}{\kern0pt}ASTAR\ cs\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}asize\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ \isanewline
-\ \ erase\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ rexp{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}erase\ AZERO\ {\isacharequal}{\kern0pt}\ ZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}erase\ {\isacharparenleft}{\kern0pt}AONE\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ ONE{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}erase\ {\isacharparenleft}{\kern0pt}ACHAR\ {\isacharunderscore}{\kern0pt}\ c{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ CH\ c{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}erase\ {\isacharparenleft}{\kern0pt}AALTs\ {\isacharunderscore}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ ZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}erase\ {\isacharparenleft}{\kern0pt}AALTs\ {\isacharunderscore}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}erase\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}r{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ ALT\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}erase\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}erase\ {\isacharparenleft}{\kern0pt}ASEQ\ {\isacharunderscore}{\kern0pt}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ SEQ\ {\isacharparenleft}{\kern0pt}erase\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}erase\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}erase\ {\isacharparenleft}{\kern0pt}ASTAR\ {\isacharunderscore}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ STAR\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ nonalt\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}nonalt\ {\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{2}}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ False{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}nonalt\ r\ {\isacharequal}{\kern0pt}\ True{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ good\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}good\ AZERO\ {\isacharequal}{\kern0pt}\ False{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}good\ {\isacharparenleft}{\kern0pt}AONE\ cs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ True{\isachardoublequoteclose}\ \isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}good\ {\isacharparenleft}{\kern0pt}ACHAR\ cs\ c{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ True{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}good\ {\isacharparenleft}{\kern0pt}AALTs\ cs\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ False{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}good\ {\isacharparenleft}{\kern0pt}AALTs\ cs\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ False{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}good\ {\isacharparenleft}{\kern0pt}AALTs\ cs\ {\isacharparenleft}{\kern0pt}r{\isadigit{1}}{\isacharhash}{\kern0pt}r{\isadigit{2}}{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymforall}r{\isacharprime}{\kern0pt}\ {\isasymin}\ set\ {\isacharparenleft}{\kern0pt}r{\isadigit{1}}{\isacharhash}{\kern0pt}r{\isadigit{2}}{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ good\ r{\isacharprime}{\kern0pt}\ {\isasymand}\ nonalt\ r{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}good\ {\isacharparenleft}{\kern0pt}ASEQ\ {\isacharunderscore}{\kern0pt}\ AZERO\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ False{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}good\ {\isacharparenleft}{\kern0pt}ASEQ\ {\isacharunderscore}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AONE\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ False{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}good\ {\isacharparenleft}{\kern0pt}ASEQ\ {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ AZERO{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ False{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}good\ {\isacharparenleft}{\kern0pt}ASEQ\ cs\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}good\ r{\isadigit{1}}\ {\isasymand}\ good\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}good\ {\isacharparenleft}{\kern0pt}ASTAR\ cs\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ True{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ fuse\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}bit\ list\ {\isasymRightarrow}\ arexp\ {\isasymRightarrow}\ arexp{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}fuse\ bs\ AZERO\ {\isacharequal}{\kern0pt}\ AZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}fuse\ bs\ {\isacharparenleft}{\kern0pt}AONE\ cs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ AONE\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ cs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}fuse\ bs\ {\isacharparenleft}{\kern0pt}ACHAR\ cs\ c{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ ACHAR\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ cs{\isacharparenright}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}fuse\ bs\ {\isacharparenleft}{\kern0pt}AALTs\ cs\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ AALTs\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ cs{\isacharparenright}{\kern0pt}\ rs{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}fuse\ bs\ {\isacharparenleft}{\kern0pt}ASEQ\ cs\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ ASEQ\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ cs{\isacharparenright}{\kern0pt}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}fuse\ bs\ {\isacharparenleft}{\kern0pt}ASTAR\ cs\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ ASTAR\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ cs{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ fuse{\isacharunderscore}{\kern0pt}append{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}fuse\ {\isacharparenleft}{\kern0pt}bs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ bs{\isadigit{2}}{\isacharparenright}{\kern0pt}\ r\ {\isacharequal}{\kern0pt}\ fuse\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{2}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ intern\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rexp\ {\isasymRightarrow}\ arexp{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}intern\ ZERO\ {\isacharequal}{\kern0pt}\ AZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}intern\ ONE\ {\isacharequal}{\kern0pt}\ AONE\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}intern\ {\isacharparenleft}{\kern0pt}CH\ c{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ ACHAR\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}intern\ {\isacharparenleft}{\kern0pt}ALT\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ AALT\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fuse\ {\isacharbrackleft}{\kern0pt}Z{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}intern\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}fuse\ {\isacharbrackleft}{\kern0pt}S{\isacharbrackright}{\kern0pt}\ \ {\isacharparenleft}{\kern0pt}intern\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}intern\ {\isacharparenleft}{\kern0pt}SEQ\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ ASEQ\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}intern\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}intern\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}intern\ {\isacharparenleft}{\kern0pt}STAR\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ ASTAR\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ retrieve\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ val\ {\isasymRightarrow}\ bit\ list{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}retrieve\ {\isacharparenleft}{\kern0pt}AONE\ bs{\isacharparenright}{\kern0pt}\ Void\ {\isacharequal}{\kern0pt}\ bs{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}retrieve\ {\isacharparenleft}{\kern0pt}ACHAR\ bs\ c{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Char\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bs{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}retrieve\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ retrieve\ r\ v{\isachardoublequoteclose}\ \isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}retrieve\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}r{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Left\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ retrieve\ r\ v{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}retrieve\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}r{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Right\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ retrieve\ {\isacharparenleft}{\kern0pt}AALTs\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}retrieve\ {\isacharparenleft}{\kern0pt}ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Seq\ v{\isadigit{1}}\ v{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ retrieve\ r{\isadigit{1}}\ v{\isadigit{1}}\ {\isacharat}{\kern0pt}\ retrieve\ r{\isadigit{2}}\ v{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}retrieve\ {\isacharparenleft}{\kern0pt}ASTAR\ bs\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Stars\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}S{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}retrieve\ {\isacharparenleft}{\kern0pt}ASTAR\ bs\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Stars\ {\isacharparenleft}{\kern0pt}v{\isacharhash}{\kern0pt}vs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ \ \ \ bs\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}Z{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ retrieve\ r\ v\ {\isacharat}{\kern0pt}\ retrieve\ {\isacharparenleft}{\kern0pt}ASTAR\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Stars\ vs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\isanewline
-\ bnullable\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}AZERO{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ False{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}AONE\ bs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ True{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}ACHAR\ bs\ c{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ False{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymexists}r\ {\isasymin}\ set\ rs{\isachardot}{\kern0pt}\ bnullable\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bnullable\ r{\isadigit{1}}\ {\isasymand}\ bnullable\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}ASTAR\ bs\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ True{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ \isanewline
-\ \ bmkeps\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ bit\ list{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}bmkeps{\isacharparenleft}{\kern0pt}AONE\ bs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bs{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bmkeps{\isacharparenleft}{\kern0pt}ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bmkeps\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bmkeps\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bmkeps{\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bmkeps\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bmkeps{\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}r{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ bnullable{\isacharparenleft}{\kern0pt}r{\isacharparenright}{\kern0pt}\ then\ bs\ {\isacharat}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bmkeps\ r{\isacharparenright}{\kern0pt}\ else\ {\isacharparenleft}{\kern0pt}bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bmkeps{\isacharparenleft}{\kern0pt}ASTAR\ bs\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}S{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\isanewline
-\ bder\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}char\ {\isasymRightarrow}\ arexp\ {\isasymRightarrow}\ arexp{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}bder\ c\ {\isacharparenleft}{\kern0pt}AZERO{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ AZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bder\ c\ {\isacharparenleft}{\kern0pt}AONE\ bs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ AZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bder\ c\ {\isacharparenleft}{\kern0pt}ACHAR\ bs\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ c\ {\isacharequal}{\kern0pt}\ d\ then\ AONE\ bs\ else\ AZERO{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bder\ c\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}bder\ c{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bder\ c\ {\isacharparenleft}{\kern0pt}ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ \ \ \ {\isacharparenleft}{\kern0pt}if\ bnullable\ r{\isadigit{1}}\isanewline
-\ \ \ \ \ \ then\ AALT\ bs\ {\isacharparenleft}{\kern0pt}ASEQ\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fuse\ {\isacharparenleft}{\kern0pt}bmkeps\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ else\ ASEQ\ bs\ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bder\ c\ {\isacharparenleft}{\kern0pt}ASTAR\ bs\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ ASEQ\ bs\ {\isacharparenleft}{\kern0pt}fuse\ {\isacharbrackleft}{\kern0pt}Z{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}ASTAR\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ \isanewline
-\ \ bders\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ string\ {\isasymRightarrow}\ arexp{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}bders\ r\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bders\ r\ {\isacharparenleft}{\kern0pt}c{\isacharhash}{\kern0pt}s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bders\ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bders{\isacharunderscore}{\kern0pt}append{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}bders\ r\ {\isacharparenleft}{\kern0pt}s{\isadigit{1}}\ {\isacharat}{\kern0pt}\ s{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bders\ {\isacharparenleft}{\kern0pt}bders\ r\ s{\isadigit{1}}{\isacharparenright}{\kern0pt}\ s{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ s{\isadigit{1}}\ arbitrary{\isacharcolon}{\kern0pt}\ r\ s{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bnullable{\isacharunderscore}{\kern0pt}correctness{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}nullable\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bnullable\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r\ rule{\isacharcolon}{\kern0pt}\ erase{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}erase\ {\isacharparenleft}{\kern0pt}fuse\ bs\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ erase\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r\ rule{\isacharcolon}{\kern0pt}\ erase{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{thm}\isamarkupfalse%
-\ Posix{\isachardot}{\kern0pt}induct\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ erase{\isacharunderscore}{\kern0pt}intern\ {\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}erase\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all\ add{\isacharcolon}{\kern0pt}\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ erase{\isacharunderscore}{\kern0pt}bder\ {\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}erase\ {\isacharparenleft}{\kern0pt}bder\ a\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ der\ a\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r\ rule{\isacharcolon}{\kern0pt}\ erase{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all\ add{\isacharcolon}{\kern0pt}\ erase{\isacharunderscore}{\kern0pt}fuse\ bnullable{\isacharunderscore}{\kern0pt}correctness{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ erase{\isacharunderscore}{\kern0pt}bders\ {\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}erase\ {\isacharparenleft}{\kern0pt}bders\ r\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ ders\ s\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ s\ arbitrary{\isacharcolon}{\kern0pt}\ r\ {\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ retrieve{\isacharunderscore}{\kern0pt}encode{\isacharunderscore}{\kern0pt}STARS{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymforall}v{\isasymin}set\ vs{\isachardot}{\kern0pt}\ {\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ r\ {\isasymand}\ code\ v\ {\isacharequal}{\kern0pt}\ retrieve\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}code\ {\isacharparenleft}{\kern0pt}Stars\ vs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ retrieve\ {\isacharparenleft}{\kern0pt}ASTAR\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Stars\ vs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ vs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ retrieve{\isacharunderscore}{\kern0pt}fuse{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}retrieve\ {\isacharparenleft}{\kern0pt}fuse\ bs\ r{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ retrieve\ r\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r\ arbitrary{\isacharcolon}{\kern0pt}\ v\ bs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ elim{\isacharcolon}{\kern0pt}\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{4}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \isacommand{defer}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ retrieve{\isacharunderscore}{\kern0pt}encode{\isacharunderscore}{\kern0pt}STARS\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ elim{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ vs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ x{\isadigit{2}}a{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ elim{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ list{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ elim{\isacharbang}{\kern0pt}{\isacharcolon}{\kern0pt}\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ retrieve{\isacharunderscore}{\kern0pt}fuse{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}retrieve\ {\isacharparenleft}{\kern0pt}fuse\ bs\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ retrieve\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all\ add{\isacharcolon}{\kern0pt}\ retrieve{\isacharunderscore}{\kern0pt}fuse{\isadigit{2}}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ retrieve{\isacharunderscore}{\kern0pt}code{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}code\ v\ {\isacharequal}{\kern0pt}\ retrieve\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ v{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ v\ r\ {\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all\ add{\isacharcolon}{\kern0pt}\ retrieve{\isacharunderscore}{\kern0pt}fuse\ retrieve{\isacharunderscore}{\kern0pt}encode{\isacharunderscore}{\kern0pt}STARS{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bnullable{\isacharunderscore}{\kern0pt}Hdbmkeps{\isacharunderscore}{\kern0pt}Hd{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}bnullable\ a{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ \ {\isachardoublequoteopen}bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}a\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bmkeps\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bmkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ bmkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ list{\isachardot}{\kern0pt}exhaust{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ r{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymnot}\ bnullable\ a{\isachardoublequoteclose}\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ \ {\isachardoublequoteopen}bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}a\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ r{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}x\ {\isasymin}\ set\ rs{\isachardoublequoteclose}\ {\isachardoublequoteopen}bnullable\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ \ r{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymnot}\ bnullable\ r{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}\ {\isasymexists}\ x\ {\isasymin}\ set\ rs{\isachardot}{\kern0pt}\ bnullable\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}retrieve\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}mkeps\ {\isacharparenleft}{\kern0pt}erase\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ retrieve\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}r\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}mkeps\ {\isacharparenleft}{\kern0pt}erase\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}r\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs\ arbitrary{\isacharcolon}{\kern0pt}\ r\ bs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bnullable{\isacharunderscore}{\kern0pt}correctness\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}correctness\ mkeps{\isacharunderscore}{\kern0pt}nullable\ retrieve{\isacharunderscore}{\kern0pt}fuse{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ retrieve{\isacharunderscore}{\kern0pt}fuse{\isadigit{2}}{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}\ insert{\isacharunderscore}{\kern0pt}iff\ list{\isachardot}{\kern0pt}exhaust\ list{\isachardot}{\kern0pt}set{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ mkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ mkeps{\isacharunderscore}{\kern0pt}nullable{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ a{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ append{\isacharunderscore}{\kern0pt}Nil{\isadigit{2}}\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}\ fuse{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ insert{\isacharunderscore}{\kern0pt}iff\ list{\isachardot}{\kern0pt}exhaust\ list{\isachardot}{\kern0pt}set{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ mkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ mkeps{\isacharunderscore}{\kern0pt}nullable\ retrieve{\isacharunderscore}{\kern0pt}fuse{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x{\isacharequal}{\kern0pt}{\isachardoublequoteopen}a{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x{\isacharequal}{\kern0pt}{\isachardoublequoteopen}bs{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule\ meta{\isacharunderscore}{\kern0pt}mp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule\ meta{\isacharunderscore}{\kern0pt}mp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ retrieve{\isacharunderscore}{\kern0pt}fuse{\isadigit{2}}{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}correctness{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}Nil{\isadigit{2}}\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse\ fuse{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ list{\isachardot}{\kern0pt}set{\isacharunderscore}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ mkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ mkeps{\isacharunderscore}{\kern0pt}nullable\ nullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}correctness{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}Nil{\isadigit{2}}\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}\ erase{\isacharunderscore}{\kern0pt}fuse\ fuse{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ list{\isachardot}{\kern0pt}set{\isacharunderscore}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ mkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ mkeps{\isacharunderscore}{\kern0pt}nullable\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ t{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymforall}r\ {\isasymin}\ set\ rs{\isachardot}{\kern0pt}\ nullable\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ bmkeps\ r\ {\isacharequal}{\kern0pt}\ retrieve\ r\ {\isacharparenleft}{\kern0pt}mkeps\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}nullable\ {\isacharparenleft}{\kern0pt}erase\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ retrieve\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}mkeps\ {\isacharparenleft}{\kern0pt}erase\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs\ arbitrary{\isacharcolon}{\kern0pt}\ bs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}correctness{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}correctness{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{2}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}assumption{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x{\isacharequal}{\kern0pt}{\isachardoublequoteopen}bs{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule\ meta{\isacharunderscore}{\kern0pt}mp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ a{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ bnullable{\isacharunderscore}{\kern0pt}Hdbmkeps{\isacharunderscore}{\kern0pt}Hd{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}nullable\ {\isacharparenleft}{\kern0pt}erase\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bnullable{\isacharunderscore}{\kern0pt}correctness\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}\ list{\isachardot}{\kern0pt}exhaust\ mkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ retrieve{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ retrieve{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ r{\isadigit{2}}\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x{\isacharequal}{\kern0pt}{\isachardoublequoteopen}bs{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule\ meta{\isacharunderscore}{\kern0pt}mp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ r{\isadigit{3}}\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ r{\isadigit{3}}\ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bmkeps{\isacharunderscore}{\kern0pt}retrieve{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}nullable\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bmkeps\ r\ {\isacharequal}{\kern0pt}\ retrieve\ r\ {\isacharparenleft}{\kern0pt}mkeps\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{defer}\isamarkupfalse%
-\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ t{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bder{\isacharunderscore}{\kern0pt}retrieve{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ der\ c\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}retrieve\ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ retrieve\ r\ {\isacharparenleft}{\kern0pt}injval\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}\ c\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r\ arbitrary{\isacharcolon}{\kern0pt}\ v\ rule{\isacharcolon}{\kern0pt}\ erase{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}c\ {\isacharequal}{\kern0pt}\ ca{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rename{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}r\isactrlsub {\isadigit{1}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}r\isactrlsub {\isadigit{2}}{\isachardoublequoteclose}\ rs\ v{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ injval{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ injval{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ retrieve{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ retrieve{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}\ same{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}nullable\ {\isacharparenleft}{\kern0pt}erase\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ r{\isadigit{1}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bnullable{\isacharunderscore}{\kern0pt}correctness\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ r{\isadigit{1}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bnullable{\isacharunderscore}{\kern0pt}correctness\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ retrieve{\isacharunderscore}{\kern0pt}fuse{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bmkeps{\isacharunderscore}{\kern0pt}retrieve{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bnullable{\isacharunderscore}{\kern0pt}correctness\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rename{\isacharunderscore}{\kern0pt}tac\ bs\ r\ v{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}clarify{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}clarify{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ injval{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ del{\isacharcolon}{\kern0pt}\ retrieve{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ retrieve{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ retrieve{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ retrieve{\isacharunderscore}{\kern0pt}fuse{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\ \ \isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ MAIN{\isacharunderscore}{\kern0pt}decode{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ ders\ s\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}Some\ {\isacharparenleft}{\kern0pt}flex\ r\ id\ s\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode\ {\isacharparenleft}{\kern0pt}retrieve\ {\isacharparenleft}{\kern0pt}bders\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induct\ s\ arbitrary{\isacharcolon}{\kern0pt}\ v\ rule{\isacharcolon}{\kern0pt}\ rev{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ Nil\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ ders\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ r{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ fact\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ r{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}Some\ v\ {\isacharequal}{\kern0pt}\ decode\ {\isacharparenleft}{\kern0pt}retrieve\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ decode{\isacharunderscore}{\kern0pt}code\ retrieve{\isacharunderscore}{\kern0pt}code\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}Some\ {\isacharparenleft}{\kern0pt}flex\ r\ id\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode\ {\isacharparenleft}{\kern0pt}retrieve\ {\isacharparenleft}{\kern0pt}bders\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}snoc\ c\ s\ v{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ IH{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}v{\isachardot}{\kern0pt}\ {\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ ders\ s\ r\ {\isasymLongrightarrow}\ \isanewline
-\ \ \ \ \ Some\ {\isacharparenleft}{\kern0pt}flex\ r\ id\ s\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode\ {\isacharparenleft}{\kern0pt}retrieve\ {\isacharparenleft}{\kern0pt}bders\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ fact\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymTurnstile}\ v\ {\isacharcolon}{\kern0pt}\ ders\ {\isacharparenleft}{\kern0pt}s\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}c{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ fact\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymTurnstile}\ injval\ {\isacharparenleft}{\kern0pt}ders\ s\ r{\isacharparenright}{\kern0pt}\ c\ v\ {\isacharcolon}{\kern0pt}\ ders\ s\ r{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Prf{\isacharunderscore}{\kern0pt}injval\ ders{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}Some\ {\isacharparenleft}{\kern0pt}flex\ r\ id\ {\isacharparenleft}{\kern0pt}s\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}c{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ Some\ {\isacharparenleft}{\kern0pt}flex\ r\ id\ s\ {\isacharparenleft}{\kern0pt}injval\ {\isacharparenleft}{\kern0pt}ders\ s\ r{\isacharparenright}{\kern0pt}\ c\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ flex{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode\ {\isacharparenleft}{\kern0pt}retrieve\ {\isacharparenleft}{\kern0pt}bders\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}injval\ {\isacharparenleft}{\kern0pt}ders\ s\ r{\isacharparenright}{\kern0pt}\ c\ v{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{2}}\ IH\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode\ {\isacharparenleft}{\kern0pt}retrieve\ {\isacharparenleft}{\kern0pt}bder\ c\ {\isacharparenleft}{\kern0pt}bders\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all\ add{\isacharcolon}{\kern0pt}\ bder{\isacharunderscore}{\kern0pt}retrieve\ ders{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}Some\ {\isacharparenleft}{\kern0pt}flex\ r\ id\ {\isacharparenleft}{\kern0pt}s\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}c{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ decode\ {\isacharparenleft}{\kern0pt}retrieve\ {\isacharparenleft}{\kern0pt}bders\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}s\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}c{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bders{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ blex\ \isakeyword{where}\isanewline
-\ {\isachardoublequoteopen}blex\ a\ s\ {\isasymequiv}\ if\ bnullable\ {\isacharparenleft}{\kern0pt}bders\ a\ s{\isacharparenright}{\kern0pt}\ then\ Some\ {\isacharparenleft}{\kern0pt}bmkeps\ {\isacharparenleft}{\kern0pt}bders\ a\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ else\ None{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ blexer\ \isakeyword{where}\isanewline
-\ {\isachardoublequoteopen}blexer\ r\ s\ {\isasymequiv}\ if\ bnullable\ {\isacharparenleft}{\kern0pt}bders\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ then\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ decode\ {\isacharparenleft}{\kern0pt}bmkeps\ {\isacharparenleft}{\kern0pt}bders\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r\ else\ None{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ blexer{\isacharunderscore}{\kern0pt}correctness{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}blexer\ r\ s\ {\isacharequal}{\kern0pt}\ lexer\ r\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{{\isacharbraceleft}{\kern0pt}}\isamarkupfalse%
-\ \isacommand{define}\isamarkupfalse%
-\ bds\ \isakeyword{where}\ {\isachardoublequoteopen}bds\ {\isasymequiv}\ bders\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{define}\isamarkupfalse%
-\ ds\ \ \isakeyword{where}\ {\isachardoublequoteopen}ds\ {\isasymequiv}\ ders\ s\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nullable\ ds{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ era{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}erase\ bds\ {\isacharequal}{\kern0pt}\ ds{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ ds{\isacharunderscore}{\kern0pt}def\ bds{\isacharunderscore}{\kern0pt}def\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ mke{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymTurnstile}\ mkeps\ ds\ {\isacharcolon}{\kern0pt}\ ds{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mkeps{\isacharunderscore}{\kern0pt}nullable{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}decode\ {\isacharparenleft}{\kern0pt}bmkeps\ bds{\isacharparenright}{\kern0pt}\ r\ {\isacharequal}{\kern0pt}\ decode\ {\isacharparenleft}{\kern0pt}retrieve\ bds\ {\isacharparenleft}{\kern0pt}mkeps\ ds{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ bmkeps{\isacharunderscore}{\kern0pt}retrieve\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ era\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bmkeps{\isacharunderscore}{\kern0pt}retrieve{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ \ Some\ {\isacharparenleft}{\kern0pt}flex\ r\ id\ s\ {\isacharparenleft}{\kern0pt}mkeps\ ds{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ mke\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all\ add{\isacharcolon}{\kern0pt}\ MAIN{\isacharunderscore}{\kern0pt}decode\ ds{\isacharunderscore}{\kern0pt}def\ bds{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}decode\ {\isacharparenleft}{\kern0pt}bmkeps\ bds{\isacharparenright}{\kern0pt}\ r\ {\isacharequal}{\kern0pt}\ Some\ {\isacharparenleft}{\kern0pt}flex\ r\ id\ s\ {\isacharparenleft}{\kern0pt}mkeps\ ds{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ bds{\isacharunderscore}{\kern0pt}def\ ds{\isacharunderscore}{\kern0pt}def\ \isacommand{{\isachardot}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{{\isacharbraceright}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}blexer\ r\ s\ {\isacharequal}{\kern0pt}\ lexer\ r\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ blexer{\isacharunderscore}{\kern0pt}def\ lexer{\isacharunderscore}{\kern0pt}flex\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ bnullable{\isacharunderscore}{\kern0pt}correctness{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{done}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ distinctBy\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ list\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}b{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}b\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ list{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}distinctBy\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ f\ acc\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}distinctBy\ {\isacharparenleft}{\kern0pt}x{\isacharhash}{\kern0pt}xs{\isacharparenright}{\kern0pt}\ f\ acc\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}f\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ acc\ then\ distinctBy\ xs\ f\ acc\ \isanewline
-\ \ \ \ \ \ else\ x\ {\isacharhash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}distinctBy\ xs\ f\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}f\ x{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ acc{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ flts\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ list\ {\isasymRightarrow}\ arexp\ list{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{where}\ \isanewline
-\ \ {\isachardoublequoteopen}flts\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}flts\ {\isacharparenleft}{\kern0pt}AZERO\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ flts\ rs{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}flts\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}AALTs\ bs\ \ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isacharparenright}{\kern0pt}\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}\ flts\ rs{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}flts\ {\isacharparenleft}{\kern0pt}r{\isadigit{1}}\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{1}}\ {\isacharhash}{\kern0pt}\ flts\ rs{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ li\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}bit\ list\ {\isasymRightarrow}\ arexp\ list\ {\isasymRightarrow}\ arexp{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}li\ {\isacharunderscore}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ AZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}li\ bs\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ fuse\ bs\ a{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}li\ bs\ as\ {\isacharequal}{\kern0pt}\ AALTs\ bs\ as{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ bsimp{\isacharunderscore}{\kern0pt}ASEQ\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}bit\ list\ {\isasymRightarrow}\ arexp\ {\isasymRightarrow}\ arexp\ {\isasymRightarrow}\ arexp{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}bsimp{\isacharunderscore}{\kern0pt}ASEQ\ {\isacharunderscore}{\kern0pt}\ AZERO\ {\isacharunderscore}{\kern0pt}\ {\isacharequal}{\kern0pt}\ AZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bsimp{\isacharunderscore}{\kern0pt}ASEQ\ {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ AZERO\ {\isacharequal}{\kern0pt}\ AZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bsimp{\isacharunderscore}{\kern0pt}ASEQ\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}AONE\ bs{\isadigit{2}}{\isacharparenright}{\kern0pt}\ r{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ fuse\ {\isacharparenleft}{\kern0pt}bs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ bs{\isadigit{2}}{\isacharparenright}{\kern0pt}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bsimp{\isacharunderscore}{\kern0pt}ASEQ\ bs{\isadigit{1}}\ r{\isadigit{1}}\ r{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ ASEQ\ \ bs{\isadigit{1}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ bsimp{\isacharunderscore}{\kern0pt}AALTs\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}bit\ list\ {\isasymRightarrow}\ arexp\ list\ {\isasymRightarrow}\ arexp{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}bsimp{\isacharunderscore}{\kern0pt}AALTs\ {\isacharunderscore}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ AZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bsimp{\isacharunderscore}{\kern0pt}AALTs\ bs{\isadigit{1}}\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ fuse\ bs{\isadigit{1}}\ r{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bsimp{\isacharunderscore}{\kern0pt}AALTs\ bs{\isadigit{1}}\ rs\ {\isacharequal}{\kern0pt}\ AALTs\ bs{\isadigit{1}}\ rs{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ bsimp\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ arexp{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}bsimp\ {\isacharparenleft}{\kern0pt}ASEQ\ bs{\isadigit{1}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}ASEQ\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}bsimp\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bsimp\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bsimp\ {\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{1}}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}AALTs\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}distinctBy\ \ {\isacharparenleft}{\kern0pt}flts\ {\isacharparenleft}{\kern0pt}map\ bsimp\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ erase\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharparenright}{\kern0pt}\ {\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bsimp\ r\ {\isacharequal}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ \isanewline
-\ \ bders{\isacharunderscore}{\kern0pt}simp\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ string\ {\isasymRightarrow}\ arexp{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}bders{\isacharunderscore}{\kern0pt}simp\ r\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}bders{\isacharunderscore}{\kern0pt}simp\ r\ {\isacharparenleft}{\kern0pt}c\ {\isacharhash}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bders{\isacharunderscore}{\kern0pt}simp\ {\isacharparenleft}{\kern0pt}bsimp\ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ s{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ blexer{\isacharunderscore}{\kern0pt}simp\ \isakeyword{where}\isanewline
-\ {\isachardoublequoteopen}blexer{\isacharunderscore}{\kern0pt}simp\ r\ s\ {\isasymequiv}\ if\ bnullable\ {\isacharparenleft}{\kern0pt}bders{\isacharunderscore}{\kern0pt}simp\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ then\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ decode\ {\isacharparenleft}{\kern0pt}bmkeps\ {\isacharparenleft}{\kern0pt}bders{\isacharunderscore}{\kern0pt}simp\ {\isacharparenleft}{\kern0pt}intern\ r{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r\ else\ None{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{export{\isacharunderscore}{\kern0pt}code}\isamarkupfalse%
-\ bders{\isacharunderscore}{\kern0pt}simp\ \isakeyword{in}\ Scala\ \isakeyword{module{\isacharunderscore}{\kern0pt}name}\ Example\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bders{\isacharunderscore}{\kern0pt}simp{\isacharunderscore}{\kern0pt}append{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bders{\isacharunderscore}{\kern0pt}simp\ r\ {\isacharparenleft}{\kern0pt}s{\isadigit{1}}\ {\isacharat}{\kern0pt}\ s{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bders{\isacharunderscore}{\kern0pt}simp\ {\isacharparenleft}{\kern0pt}bders{\isacharunderscore}{\kern0pt}simp\ r\ s{\isadigit{1}}{\isacharparenright}{\kern0pt}\ s{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ s{\isadigit{1}}\ arbitrary{\isacharcolon}{\kern0pt}\ r\ s{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}bsimp{\isacharunderscore}{\kern0pt}ASEQ{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}L\ {\isacharparenleft}{\kern0pt}SEQ\ {\isacharparenleft}{\kern0pt}erase\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}erase\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ L\ {\isacharparenleft}{\kern0pt}erase\ {\isacharparenleft}{\kern0pt}bsimp{\isacharunderscore}{\kern0pt}ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}\ rule{\isacharcolon}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}ASEQ{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ erase{\isacharunderscore}{\kern0pt}fuse\ fuse{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}bsimp{\isacharunderscore}{\kern0pt}AALTs{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}L\ {\isacharparenleft}{\kern0pt}erase\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ L\ {\isacharparenleft}{\kern0pt}erase\ {\isacharparenleft}{\kern0pt}bsimp{\isacharunderscore}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ bs\ rs\ rule{\isacharcolon}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}AALTs{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all\ add{\isacharcolon}{\kern0pt}\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}AALTs{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}L\ {\isacharparenleft}{\kern0pt}erase\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymUnion}\ {\isacharparenleft}{\kern0pt}L\ {\isacharbackquote}{\kern0pt}\ erase\ {\isacharbackquote}{\kern0pt}\ {\isacharparenleft}{\kern0pt}set\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}flts{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymUnion}\ {\isacharparenleft}{\kern0pt}L\ {\isacharbackquote}{\kern0pt}\ erase\ {\isacharbackquote}{\kern0pt}\ {\isacharparenleft}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}flts\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymUnion}\ {\isacharparenleft}{\kern0pt}L\ {\isacharbackquote}{\kern0pt}\ erase\ {\isacharbackquote}{\kern0pt}\ {\isacharparenleft}{\kern0pt}set\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs\ rule{\isacharcolon}{\kern0pt}\ flts{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}AALTs\ erase{\isacharunderscore}{\kern0pt}fuse\ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}AALTs\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}dB{\isacharunderscore}{\kern0pt}acc{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}\ {\isasymUnion}{\isacharparenleft}{\kern0pt}L\ {\isacharbackquote}{\kern0pt}\ acc{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}\ {\isasymUnion}\ {\isacharparenleft}{\kern0pt}L\ {\isacharbackquote}{\kern0pt}\ erase\ {\isacharbackquote}{\kern0pt}\ {\isacharparenleft}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}distinctBy\ rs\ erase\ acc{\isacharparenright}{\kern0pt}\ {\isacharparenright}{\kern0pt}\ {\isacharparenright}{\kern0pt}\ {\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymUnion}{\isacharparenleft}{\kern0pt}L\ {\isacharbackquote}{\kern0pt}\ acc{\isacharparenright}{\kern0pt}\ {\isasymunion}\ \ {\isasymUnion}\ {\isacharparenleft}{\kern0pt}L\ {\isacharbackquote}{\kern0pt}\ erase\ {\isacharbackquote}{\kern0pt}\ {\isacharparenleft}{\kern0pt}set\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ rs\ arbitrary{\isacharcolon}{\kern0pt}\ acc{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}z{\isadigit{3}}{\isacharparenright}{\kern0pt}\ SUP{\isacharunderscore}{\kern0pt}absorb\ UN{\isacharunderscore}{\kern0pt}insert\ sup{\isacharunderscore}{\kern0pt}assoc\ sup{\isacharunderscore}{\kern0pt}commute{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}dB{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}\ {\isacharparenleft}{\kern0pt}\ {\isasymUnion}\ {\isacharparenleft}{\kern0pt}L\ {\isacharbackquote}{\kern0pt}\ erase\ {\isacharbackquote}{\kern0pt}\ {\isacharparenleft}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}distinctBy\ rs\ erase\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenright}{\kern0pt}\ {\isacharparenright}{\kern0pt}\ {\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isasymUnion}\ {\isacharparenleft}{\kern0pt}L\ {\isacharbackquote}{\kern0pt}\ erase\ {\isacharbackquote}{\kern0pt}\ {\isacharparenleft}{\kern0pt}set\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}dB{\isacharunderscore}{\kern0pt}acc\ Un{\isacharunderscore}{\kern0pt}commute\ Union{\isacharunderscore}{\kern0pt}image{\isacharunderscore}{\kern0pt}empty{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}bsimp{\isacharunderscore}{\kern0pt}erase{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}L\ {\isacharparenleft}{\kern0pt}erase\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ L\ {\isacharparenleft}{\kern0pt}erase\ {\isacharparenleft}{\kern0pt}bsimp\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ Sequ{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ L{\isacharunderscore}{\kern0pt}bsimp{\isacharunderscore}{\kern0pt}ASEQ{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ Sequ{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ {\isacharparenleft}{\kern0pt}asm{\isacharparenright}{\kern0pt}\ \ L{\isacharunderscore}{\kern0pt}bsimp{\isacharunderscore}{\kern0pt}ASEQ{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto\ simp\ add{\isacharcolon}{\kern0pt}\ Sequ{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ L{\isacharunderscore}{\kern0pt}bsimp{\isacharunderscore}{\kern0pt}AALTs{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{defer}\isamarkupfalse%
-\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ {\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}AALTs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}dB{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}flts{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}AALTs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}AALTs\ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bsimp{\isacharunderscore}{\kern0pt}ASEQ{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bsimp{\isacharunderscore}{\kern0pt}ASEQ\ bs\ r{\isadigit{1}}\ AZERO\ {\isacharequal}{\kern0pt}\ AZERO{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bsimp{\isacharunderscore}{\kern0pt}ASEQ{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}r{\isadigit{1}}\ {\isasymnoteq}\ AZERO{\isachardoublequoteclose}\ {\isachardoublequoteopen}r{\isadigit{2}}\ {\isasymnoteq}\ AZERO{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymforall}bs{\isachardot}{\kern0pt}\ r{\isadigit{1}}\ {\isasymnoteq}\ AONE\ bs{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bsimp{\isacharunderscore}{\kern0pt}ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}\ rule{\isacharcolon}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}ASEQ{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bsimp{\isacharunderscore}{\kern0pt}ASEQ{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bsimp{\isacharunderscore}{\kern0pt}ASEQ\ bs\ {\isacharparenleft}{\kern0pt}AONE\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ r{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ fuse\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ L{\isacharunderscore}{\kern0pt}bders{\isacharunderscore}{\kern0pt}simp{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}L\ {\isacharparenleft}{\kern0pt}erase\ {\isacharparenleft}{\kern0pt}bders{\isacharunderscore}{\kern0pt}simp\ r\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ L\ {\isacharparenleft}{\kern0pt}erase\ {\isacharparenleft}{\kern0pt}bders\ r\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ s\ arbitrary{\isacharcolon}{\kern0pt}\ r\ rule{\isacharcolon}{\kern0pt}\ rev{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ ders{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bders{\isacharunderscore}{\kern0pt}simp{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ L{\isacharunderscore}{\kern0pt}bsimp{\isacharunderscore}{\kern0pt}erase{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ der{\isacharunderscore}{\kern0pt}correctness{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ b{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}bnullable\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bmkeps\ {\isacharparenleft}{\kern0pt}fuse\ bs\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ bmkeps\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ assms\ bmkeps{\isacharunderscore}{\kern0pt}retrieve\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse\ mkeps{\isacharunderscore}{\kern0pt}nullable\ retrieve{\isacharunderscore}{\kern0pt}fuse{\isadigit{2}}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ b{\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}bders{\isacharunderscore}{\kern0pt}simp\ r\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bnullable\ {\isacharparenleft}{\kern0pt}bders\ r\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ L{\isacharunderscore}{\kern0pt}bders{\isacharunderscore}{\kern0pt}simp\ bnullable{\isacharunderscore}{\kern0pt}correctness\ lexer{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ lexer{\isacharunderscore}{\kern0pt}correct{\isacharunderscore}{\kern0pt}None\ option{\isachardot}{\kern0pt}distinct{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ qq{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymexists}r\ {\isasymin}\ set\ rs{\isachardot}{\kern0pt}\ bnullable\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs\ {\isacharat}{\kern0pt}\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs\ arbitrary{\isacharcolon}{\kern0pt}\ rs{\isadigit{1}}\ bs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ Nil{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}conv\ bmkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ neq{\isacharunderscore}{\kern0pt}Nil{\isacharunderscore}{\kern0pt}conv\ bnullable{\isacharunderscore}{\kern0pt}Hdbmkeps{\isacharunderscore}{\kern0pt}Hd\ split{\isacharunderscore}{\kern0pt}list{\isacharunderscore}{\kern0pt}last{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ qq{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymforall}r\ {\isasymin}\ set\ rs{\isachardot}{\kern0pt}\ {\isasymnot}\ bnullable\ r{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isasymexists}r\ {\isasymin}\ set\ rs{\isadigit{1}}{\isachardot}{\kern0pt}\ bnullable\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs\ {\isacharat}{\kern0pt}\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs\ arbitrary{\isacharcolon}{\kern0pt}\ rs{\isadigit{1}}\ bs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}assoc\ in{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}conv{\isacharunderscore}{\kern0pt}decomp\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\ \ \isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ qq{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymexists}r\ {\isasymin}\ set\ rs{\isachardot}{\kern0pt}\ bnullable\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs\ arbitrary{\isacharcolon}{\kern0pt}\ bs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ nonnested\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}nonnested\ {\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{2}}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ True{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}nonnested\ {\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{2}}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{1}}\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharhash}{\kern0pt}\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ False{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}nonnested\ {\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{2}}\ {\isacharparenleft}{\kern0pt}r\ {\isacharhash}{\kern0pt}\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ nonnested\ {\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{2}}\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}nonnested\ r\ {\isacharequal}{\kern0pt}\ True{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ \ k{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}flts\ {\isacharparenleft}{\kern0pt}r\ {\isacharhash}{\kern0pt}\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ flts\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ flts\ rs{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r\ arbitrary{\isacharcolon}{\kern0pt}\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ \ k{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}flts\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ flts\ rs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ flts\ rs{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs{\isadigit{1}}\ arbitrary{\isacharcolon}{\kern0pt}\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isachardot}{\kern0pt}assoc\ k{\isadigit{0}}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ \ k{\isadigit{0}}a{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}flts\ {\isacharbrackleft}{\kern0pt}AALTs\ bs\ rs{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isacharparenright}{\kern0pt}\ \ rs{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bsimp{\isacharunderscore}{\kern0pt}AALTs{\isacharunderscore}{\kern0pt}qq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isadigit{1}}\ {\isacharless}{\kern0pt}\ length\ rs{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bsimp{\isacharunderscore}{\kern0pt}AALTs\ bs\ rs\ {\isacharequal}{\kern0pt}\ AALTs\ bs\ \ rs{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ \ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ list{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bbbbs{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}nonalt\ r\ {\isasymor}\ {\isacharparenleft}{\kern0pt}{\isasymexists}bs\ rs{\isachardot}{\kern0pt}\ r\ \ {\isacharequal}{\kern0pt}\ AALTs\ bs\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ nonalt{\isachardot}{\kern0pt}elims{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\ \ \isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ flts{\isacharunderscore}{\kern0pt}append{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}flts\ {\isacharparenleft}{\kern0pt}xs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ xs{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ flts\ xs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ flts\ xs{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ xs{\isadigit{1}}\ \ arbitrary{\isacharcolon}{\kern0pt}\ xs{\isadigit{2}}\ \ rule{\isacharcolon}{\kern0pt}\ rev{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ xs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ nonazero\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}nonazero\ AZERO\ {\isacharequal}{\kern0pt}\ False{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}nonazero\ r\ {\isacharequal}{\kern0pt}\ True{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ flts{\isacharunderscore}{\kern0pt}single{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}nonalt\ r{\isachardoublequoteclose}\ {\isachardoublequoteopen}nonazero\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}flts\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ q{\isadigit{3}}a{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymexists}r\ {\isasymin}\ set\ rs{\isachardot}{\kern0pt}\ bnullable\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ {\isacharparenleft}{\kern0pt}bs{\isacharat}{\kern0pt}bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs\ arbitrary{\isacharcolon}{\kern0pt}\ bs\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}assoc\ b{\isadigit{2}}\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse\ bnullable{\isacharunderscore}{\kern0pt}Hdbmkeps{\isacharunderscore}{\kern0pt}Hd{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ a{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isachardot}{\kern0pt}assoc\ b{\isadigit{2}}\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse\ bnullable{\isacharunderscore}{\kern0pt}Hdbmkeps{\isacharunderscore}{\kern0pt}Hd{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ qq{\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymexists}x{\isasymin}set\ list{\isachardot}{\kern0pt}\ bnullable\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymexists}x{\isasymin}set\ {\isacharparenleft}{\kern0pt}flts\ list{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ bnullable\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ list\ rule{\isacharcolon}{\kern0pt}\ flts{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ UnCI\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse\ imageI{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\ \ \isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ qs{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymexists}r\ {\isasymin}\ set\ rs{\isachardot}{\kern0pt}\ bnullable\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}flts\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs\ arbitrary{\isacharcolon}{\kern0pt}\ bs\ taking{\isacharcolon}{\kern0pt}\ size\ rule{\isacharcolon}{\kern0pt}\ measure{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ x{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ a{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}Hdbmkeps{\isacharunderscore}{\kern0pt}Hd{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}flts\ list{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}AALTs\ L{\isacharunderscore}{\kern0pt}erase{\isacharunderscore}{\kern0pt}flts\ L{\isacharunderscore}{\kern0pt}flat{\isacharunderscore}{\kern0pt}Prf{\isadigit{1}}\ L{\isacharunderscore}{\kern0pt}flat{\isacharunderscore}{\kern0pt}Prf{\isadigit{2}}\ Prf{\isacharunderscore}{\kern0pt}elims{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ mkeps{\isacharunderscore}{\kern0pt}nullable\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}Hdbmkeps{\isacharunderscore}{\kern0pt}Hd{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}{\isasymexists}x{\isasymin}set\ x{\isadigit{5}}{\isadigit{2}}{\isachardot}{\kern0pt}\ bnullable\ x{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}list{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ b{\isadigit{2}}\ fuse{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ q{\isadigit{3}}a\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ disjE{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ qq{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ b{\isadigit{2}}\ fuse{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ q{\isadigit{3}}a\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ qq{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse\ image{\isacharunderscore}{\kern0pt}eqI\ set{\isacharunderscore}{\kern0pt}map{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ b{\isadigit{2}}\ fuse{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ q{\isadigit{3}}a\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ qq{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse\ image{\isacharunderscore}{\kern0pt}eqI\ set{\isacharunderscore}{\kern0pt}map{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ b{\isadigit{2}}\ fuse{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ q{\isadigit{3}}a\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ qq{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse\ imageE\ set{\isacharunderscore}{\kern0pt}map{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}list{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ qq{\isadigit{4}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ list{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}Hdbmkeps{\isacharunderscore}{\kern0pt}Hd{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}ASEQ\ x{\isadigit{4}}{\isadigit{1}}\ x{\isadigit{4}}{\isadigit{2}}\ x{\isadigit{4}}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ list{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}Hdbmkeps{\isacharunderscore}{\kern0pt}Hd{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ qq{\isadigit{4}}\ r{\isadigit{1}}\ r{\isadigit{2}}\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\ \ \isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bder{\isacharunderscore}{\kern0pt}fuse{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bder\ c\ {\isacharparenleft}{\kern0pt}fuse\ bs\ a{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ fuse\ bs\ \ {\isacharparenleft}{\kern0pt}bder\ c\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ a\ arbitrary{\isacharcolon}{\kern0pt}\ bs\ c{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ flts{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}char\ {\isasymRightarrow}\ arexp\ list\ {\isasymRightarrow}\ arexp\ list{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{where}\ \isanewline
-\ \ {\isachardoublequoteopen}flts{\isadigit{2}}\ {\isacharunderscore}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}flts{\isadigit{2}}\ c\ {\isacharparenleft}{\kern0pt}AZERO\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ flts{\isadigit{2}}\ c\ rs{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}flts{\isadigit{2}}\ c\ {\isacharparenleft}{\kern0pt}AONE\ {\isacharunderscore}{\kern0pt}\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ flts{\isadigit{2}}\ c\ rs{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}flts{\isadigit{2}}\ c\ {\isacharparenleft}{\kern0pt}ACHAR\ bs\ d\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ c\ {\isacharequal}{\kern0pt}\ d\ then\ {\isacharparenleft}{\kern0pt}ACHAR\ bs\ d\ {\isacharhash}{\kern0pt}\ flts{\isadigit{2}}\ c\ rs{\isacharparenright}{\kern0pt}\ else\ flts{\isadigit{2}}\ c\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}flts{\isadigit{2}}\ c\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isacharparenright}{\kern0pt}\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}\ flts{\isadigit{2}}\ c\ rs{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}flts{\isadigit{2}}\ c\ {\isacharparenleft}{\kern0pt}ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}bnullable{\isacharparenleft}{\kern0pt}r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ r{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ AZERO{\isacharparenright}{\kern0pt}\ then\ \isanewline
-\ \ \ \ flts{\isadigit{2}}\ c\ rs\isanewline
-\ \ \ \ else\ ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}\ {\isacharhash}{\kern0pt}\ flts{\isadigit{2}}\ c\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}flts{\isadigit{2}}\ c\ {\isacharparenleft}{\kern0pt}r{\isadigit{1}}\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{1}}\ {\isacharhash}{\kern0pt}\ flts{\isadigit{2}}\ c\ rs{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\ \isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ WQ{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}s\ {\isasymin}\ L\ {\isacharparenleft}{\kern0pt}der\ c\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}s\ {\isasymin}\ der\ c\ r\ {\isasymrightarrow}\ mkeps\ {\isacharparenleft}{\kern0pt}ders\ s\ {\isacharparenleft}{\kern0pt}der\ c\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{oops}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bder{\isacharunderscore}{\kern0pt}bsimp{\isacharunderscore}{\kern0pt}AALTs{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bder\ c\ {\isacharparenleft}{\kern0pt}bsimp{\isacharunderscore}{\kern0pt}AALTs\ bs\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}bder\ c{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ bs\ rs\ rule{\isacharcolon}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}AALTs{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\ \ \isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bder{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}asize\ {\isacharparenleft}{\kern0pt}bsimp\ a{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ asize\ a{\isachardoublequoteclose}\ \ {\isachardoublequoteopen}a\ {\isacharequal}{\kern0pt}\ AALTs\ bs\ {\isacharbrackleft}{\kern0pt}AALTs\ bs{\isadigit{2}}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharcomma}{\kern0pt}\ AZERO{\isacharcomma}{\kern0pt}\ AONE\ bs{\isadigit{3}}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}bsimp\ a\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{oops}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{inductive}\isamarkupfalse%
-\ rrewrite{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ arexp\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}\ {\isasymleadsto}\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbrackleft}{\kern0pt}{\isadigit{9}}{\isadigit{9}}{\isacharcomma}{\kern0pt}\ {\isadigit{9}}{\isadigit{9}}{\isacharbrackright}{\kern0pt}\ {\isadigit{9}}{\isadigit{9}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}ASEQ\ bs\ AZERO\ r{\isadigit{2}}\ {\isasymleadsto}\ AZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}ASEQ\ bs\ r{\isadigit{1}}\ AZERO\ {\isasymleadsto}\ AZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}ASEQ\ bs\ {\isacharparenleft}{\kern0pt}AONE\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ r\ {\isasymleadsto}\ fuse\ {\isacharparenleft}{\kern0pt}bs{\isacharat}{\kern0pt}bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}r{\isadigit{1}}\ {\isasymleadsto}\ r{\isadigit{2}}\ {\isasymLongrightarrow}\ ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{3}}\ {\isasymleadsto}\ ASEQ\ bs\ r{\isadigit{2}}\ r{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}r{\isadigit{3}}\ {\isasymleadsto}\ r{\isadigit{4}}\ {\isasymLongrightarrow}\ ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{3}}\ {\isasymleadsto}\ ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{4}}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}r\ {\isasymleadsto}\ r{\isacharprime}{\kern0pt}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}r{\isacharprime}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa{\isacharat}{\kern0pt}AZERO\ {\isacharhash}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa{\isacharat}{\kern0pt}rsb{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa{\isacharat}{\kern0pt}{\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{1}}\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharhash}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa{\isacharat}{\kern0pt}{\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharat}{\kern0pt}rsb{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}AALTs\ bs\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ {\isacharparenleft}{\kern0pt}bs{\isacharat}{\kern0pt}bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isachardoublequoteclose}\isanewline
-\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}AALTs\ {\isacharparenleft}{\kern0pt}bs{\isacharat}{\kern0pt}bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}AALTs\ bs\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymleadsto}\ AZERO{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}AALTs\ bs\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}\ {\isasymleadsto}\ fuse\ bs\ r{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}erase\ a{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ erase\ a{\isadigit{2}}\ {\isasymLongrightarrow}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isadigit{1}}{\isacharbrackright}{\kern0pt}{\isacharat}{\kern0pt}rsb{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isadigit{2}}{\isacharbrackright}{\kern0pt}{\isacharat}{\kern0pt}rsc{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isadigit{1}}{\isacharbrackright}{\kern0pt}{\isacharat}{\kern0pt}rsb{\isacharat}{\kern0pt}rsc{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{inductive}\isamarkupfalse%
-\ rrewrites{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ {\isasymRightarrow}\ arexp\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharcomma}{\kern0pt}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharbrackright}{\kern0pt}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isakeyword{where}\ \isanewline
-rs{\isadigit{1}}{\isacharbrackleft}{\kern0pt}intro{\isacharcomma}{\kern0pt}\ simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}{\isachardoublequoteopen}r\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ rs{\isadigit{2}}{\isacharbrackleft}{\kern0pt}intro{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}r{\isadigit{1}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{2}}{\isacharsemicolon}{\kern0pt}\ r{\isadigit{2}}\ {\isasymleadsto}\ r{\isadigit{3}}{\isasymrbrakk}\ {\isasymLongrightarrow}\ r{\isadigit{1}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{inductive}\isamarkupfalse%
-\ srewrites{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ list\ {\isasymRightarrow}\ arexp\ list\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}\ {\isacharunderscore}{\kern0pt}\ s{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharcomma}{\kern0pt}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharbrackright}{\kern0pt}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isakeyword{where}\isanewline
-ss{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ s{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}ss{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}r\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isacharprime}{\kern0pt}{\isacharsemicolon}{\kern0pt}\ rs\ s{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ rs{\isacharprime}{\kern0pt}{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}r{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}\ s{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharprime}{\kern0pt}{\isacharhash}{\kern0pt}rs{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ {\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}r{\isadigit{1}}\ {\isasymleadsto}\ r{\isadigit{2}}\ {\isasymLongrightarrow}\ r{\isadigit{1}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ rrewrites{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\ \isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ real{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{assumes}\ a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}r{\isadigit{1}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{2}}{\isachardoublequoteclose}\ \ \isakeyword{and}\ a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}r{\isadigit{2}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}r{\isadigit{1}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ a{\isadigit{2}}\ a{\isadigit{1}}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r{\isadigit{2}}\ r{\isadigit{3}}\ arbitrary{\isacharcolon}{\kern0pt}\ r{\isadigit{1}}\ rule{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ \ many{\isacharunderscore}{\kern0pt}steps{\isacharunderscore}{\kern0pt}later{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}r{\isadigit{1}}\ {\isasymleadsto}\ r{\isadigit{2}}{\isacharsemicolon}{\kern0pt}\ r{\isadigit{2}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{3}}\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ r{\isadigit{1}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ real{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ contextrewrites{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}r\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isacharprime}{\kern0pt}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}r{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}r{\isacharprime}{\kern0pt}{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r\ r{\isacharprime}{\kern0pt}\ rule{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}Cons\ append{\isacharunderscore}{\kern0pt}Nil\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ contextrewrites{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}r\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isacharprime}{\kern0pt}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isacharat}{\kern0pt}rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}r{\isacharprime}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharat}{\kern0pt}rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r\ r{\isacharprime}{\kern0pt}\ rule{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ srewrites{\isacharunderscore}{\kern0pt}alt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rs{\isadigit{1}}\ s{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ rs{\isadigit{2}}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isacharat}{\kern0pt}rs{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isacharat}{\kern0pt}rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\isanewline
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs{\isadigit{1}}\ rs{\isadigit{2}}\ arbitrary{\isacharcolon}{\kern0pt}\ bs\ rs\ rule{\isacharcolon}{\kern0pt}\ srewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}bs{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}rsa{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}r{\isacharprime}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ real{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}assumption{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule\ contextrewrites{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{corollary}\isamarkupfalse%
-\ srewrites{\isacharunderscore}{\kern0pt}alt{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rs{\isadigit{1}}\ s{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ rs{\isadigit{2}}\ {\isasymLongrightarrow}\ AALTs\ bs\ rs{\isadigit{1}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ AALTs\ bs\ rs{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isachardot}{\kern0pt}left{\isacharunderscore}{\kern0pt}neutral\ srewrites{\isacharunderscore}{\kern0pt}alt{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ star{\isacharunderscore}{\kern0pt}seq{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}r{\isadigit{1}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{2}}\ {\isasymLongrightarrow}\ ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{3}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ ASEQ\ bs\ r{\isadigit{2}}\ r{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r{\isadigit{1}}\ r{\isadigit{2}}\ arbitrary{\isacharcolon}{\kern0pt}\ r{\isadigit{3}}\ rule{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ rrewrites{\isachardot}{\kern0pt}cases{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}assumption{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ assumption%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ star{\isacharunderscore}{\kern0pt}seq{\isadigit{2}}{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}r{\isadigit{3}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{4}}\ {\isasymLongrightarrow}\ ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{3}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ ASEQ\ bs\ r{\isadigit{1}}\ r{\isadigit{4}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ r{\isadigit{3}}\ r{\isadigit{4}}\ arbitrary{\isacharcolon}{\kern0pt}\ r{\isadigit{1}}\ rule{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ continuous{\isacharunderscore}{\kern0pt}rewrite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}r{\isadigit{1}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ AZERO{\isasymrbrakk}\ {\isasymLongrightarrow}\ ASEQ\ bs{\isadigit{1}}\ r{\isadigit{1}}\ r{\isadigit{2}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ AZERO{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ ra{\isasymequiv}{\isachardoublequoteopen}r{\isadigit{1}}{\isachardoublequoteclose}\ rb{\isasymequiv}{\isachardoublequoteopen}AZERO{\isachardoublequoteclose}\ arbitrary{\isacharcolon}{\kern0pt}\ bs{\isadigit{1}}\ r{\isadigit{1}}\ r{\isadigit{2}}\ rule{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ star{\isacharunderscore}{\kern0pt}seq{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\ \ \isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bsimp{\isacharunderscore}{\kern0pt}aalts{\isacharunderscore}{\kern0pt}simpcases{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}AONE\ bs\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bsimp\ {\isacharparenleft}{\kern0pt}AONE\ bs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \ {\isachardoublequoteopen}AZERO\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ bsimp\ AZERO{\isachardoublequoteclose}\ {\isachardoublequoteopen}ACHAR\ bs\ c\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bsimp\ {\isacharparenleft}{\kern0pt}ACHAR\ bs\ c{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ trivialbsimpsrewrites{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}{\isasymAnd}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ set\ rs\ {\isasymLongrightarrow}\ x\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ f\ x\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ rs\ s{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}map\ f\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\isanewline
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ ss{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ insert{\isacharunderscore}{\kern0pt}iff\ list{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{5}}{\isacharparenright}{\kern0pt}\ list{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{9}}{\isacharparenright}{\kern0pt}\ srewrites{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bsimp{\isacharunderscore}{\kern0pt}AALTsrewrites{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}AALTs\ bs{\isadigit{1}}\ rs\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}AALTs\ bs{\isadigit{1}}\ rs{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ \ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}rs\ {\isacharequal}{\kern0pt}\ Nil{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}length\ {\isacharparenleft}{\kern0pt}a{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}AALTs{\isacharunderscore}{\kern0pt}qq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\ \isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{inductive}\isamarkupfalse%
-\ frewrites{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}arexp\ list\ {\isasymRightarrow}\ arexp\ list\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}\ {\isacharunderscore}{\kern0pt}\ f{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharcomma}{\kern0pt}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharbrackright}{\kern0pt}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isakeyword{where}\isanewline
-fs{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ f{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}fs{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}rs\ f{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ rs{\isacharprime}{\kern0pt}{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}AZERO{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}\ f{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ rs{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}fs{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}rs\ f{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ rs{\isacharprime}{\kern0pt}{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ f{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isacharparenright}{\kern0pt}\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}fs{\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}rs\ f{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ rs{\isacharprime}{\kern0pt}{\isacharsemicolon}{\kern0pt}nonalt\ r{\isacharsemicolon}{\kern0pt}\ nonazero\ r{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}r{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}\ f{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r{\isacharhash}{\kern0pt}rs{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ flts{\isacharunderscore}{\kern0pt}prepend{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}nonalt\ a{\isacharsemicolon}{\kern0pt}\ nonazero\ a{\isasymrbrakk}\ {\isasymLongrightarrow}\ flts\ {\isacharparenleft}{\kern0pt}a{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ a\ {\isacharhash}{\kern0pt}\ {\isacharparenleft}{\kern0pt}flts\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}Cons\ append{\isacharunderscore}{\kern0pt}Nil\ flts{\isacharunderscore}{\kern0pt}single{\isadigit{1}}\ k{\isadigit{0}}{\isadigit{0}}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ fltsfrewrites{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rs\ f{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}flts\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ fs{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}a\ {\isacharequal}{\kern0pt}\ AZERO{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ fs{\isadigit{2}}\ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}{\isasymexists}bs\ rs{\isachardot}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ AALTs\ bs\ rs{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ exE{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \ \isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ fs{\isadigit{3}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ flts{\isacharunderscore}{\kern0pt}prepend{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ nonalt{\isachardot}{\kern0pt}elims{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \isacommand{thm}\isamarkupfalse%
-\ nonalt{\isachardot}{\kern0pt}elims\isanewline
-\ \ \ \isanewline
-\ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bbbbs{\isadigit{1}}\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ nonalt{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \ \isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ nonazero{\isachardot}{\kern0pt}elims{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ fs{\isadigit{4}}\ nonalt{\isachardot}{\kern0pt}elims{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ nonazero{\isachardot}{\kern0pt}elims{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rrewrite{\isadigit{0}}away{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}AALTs\ bs\ {\isacharparenleft}{\kern0pt}\ AZERO\ {\isacharhash}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ rsb{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}Nil\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{7}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ frewritesaalts{\isacharcolon}{\kern0pt}{\isachardoublequoteopen}rs\ f{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ rs{\isacharprime}{\kern0pt}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}rs{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ rs\ rs{\isacharprime}{\kern0pt}\ arbitrary{\isacharcolon}{\kern0pt}\ bs\ rs{\isadigit{1}}\ rule{\isacharcolon}{\kern0pt}frewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}bs{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}rs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}AZERO{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ real{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{7}}{\isacharparenright}{\kern0pt}\ \isacommand{apply}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}bsa{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}rs{\isadigit{1}}a\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{1}}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ real{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{8}}{\isacharparenright}{\kern0pt}\ \isacommand{apply}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}bs{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}rs{\isadigit{1}}{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ real{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ fltsrewrites{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ \ AALTs\ bs{\isadigit{1}}\ rs\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ AALTs\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}flts\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}a\ {\isacharequal}{\kern0pt}\ AZERO{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}Nil\ flts{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ many{\isacharunderscore}{\kern0pt}steps{\isacharunderscore}{\kern0pt}later\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{7}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}{\isasymexists}bs{\isadigit{2}}\ rs{\isadigit{2}}{\isachardot}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ AALTs\ bs{\isadigit{2}}\ rs{\isadigit{2}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ exE{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ flts{\isachardot}{\kern0pt}simps{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ flts{\isacharunderscore}{\kern0pt}prepend{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ nonalt{\isachardot}{\kern0pt}elims{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ nonazero{\isachardot}{\kern0pt}elims{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}\ f{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharhash}{\kern0pt}flts\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}Nil\ frewritesaalts{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ fltsfrewrites\ fs{\isadigit{4}}\ nonalt{\isachardot}{\kern0pt}elims{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ nonazero{\isachardot}{\kern0pt}elims{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}Cons\ append{\isacharunderscore}{\kern0pt}Nil\ fltsfrewrites\ frewritesaalts\ k{\isadigit{0}}{\isadigit{0}}\ k{\isadigit{0}}a{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ alts{\isacharunderscore}{\kern0pt}simpalts{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}bs{\isadigit{1}}\ rs{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymAnd}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ set\ rs\ {\isasymLongrightarrow}\ x\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ bsimp\ x{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\ \isanewline
-AALTs\ bs{\isadigit{1}}\ rs\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ AALTs\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}map\ bsimp\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}\ rs\ s{\isasymleadsto}{\isacharasterisk}{\kern0pt}\ \ {\isacharparenleft}{\kern0pt}map\ bsimp\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ trivialbsimpsrewrites\ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ srewrites{\isacharunderscore}{\kern0pt}alt{\isadigit{1}}\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ threelistsappend{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rsa{\isacharat}{\kern0pt}a{\isacharhash}{\kern0pt}rsb\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rsa{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharat}{\kern0pt}rsb{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ distinctByAcc\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ list\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}b{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}b\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}b\ set{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}distinctByAcc\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ f\ acc\ {\isacharequal}{\kern0pt}\ acc{\isachardoublequoteclose}\isanewline
-{\isacharbar}{\kern0pt}\ {\isachardoublequoteopen}distinctByAcc\ {\isacharparenleft}{\kern0pt}x{\isacharhash}{\kern0pt}xs{\isacharparenright}{\kern0pt}\ f\ acc\ {\isacharequal}{\kern0pt}\ \isanewline
-\ \ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}f\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ acc\ then\ distinctByAcc\ xs\ f\ acc\ \isanewline
-\ \ \ \ \ \ else\ \ {\isacharparenleft}{\kern0pt}distinctByAcc\ xs\ f\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}f\ x{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ acc{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ dB{\isacharunderscore}{\kern0pt}single{\isacharunderscore}{\kern0pt}step{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}distinctBy\ {\isacharparenleft}{\kern0pt}a{\isacharhash}{\kern0pt}rs{\isacharparenright}{\kern0pt}\ f\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ a\ {\isacharhash}{\kern0pt}\ distinctBy\ rs\ f\ {\isacharbraceleft}{\kern0pt}f\ a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ somewhereInside{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}r\ {\isasymin}\ set\ rs\ {\isasymLongrightarrow}\ {\isasymexists}rs{\isadigit{1}}\ rs{\isadigit{2}}{\isachardot}{\kern0pt}\ rs\ {\isacharequal}{\kern0pt}\ rs{\isadigit{1}}{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isacharat}{\kern0pt}rs{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ split{\isacharunderscore}{\kern0pt}list\ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ somewhereMapInside{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}f\ r\ {\isasymin}\ f\ {\isacharbackquote}{\kern0pt}\ set\ rs\ {\isasymLongrightarrow}\ {\isasymexists}rs{\isadigit{1}}\ rs{\isadigit{2}}\ a{\isachardot}{\kern0pt}\ rs\ {\isacharequal}{\kern0pt}\ rs{\isadigit{1}}{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharat}{\kern0pt}rs{\isadigit{2}}\ {\isasymand}\ f\ a\ {\isacharequal}{\kern0pt}\ f\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ split{\isacharunderscore}{\kern0pt}list{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ alts{\isacharunderscore}{\kern0pt}dBrewrites{\isacharunderscore}{\kern0pt}withFront{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ distinctBy\ rs\ erase\ {\isacharparenleft}{\kern0pt}erase\ {\isacharbackquote}{\kern0pt}\ set\ rsa{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ rs\ arbitrary{\isacharcolon}{\kern0pt}\ rsa{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule{\isacharunderscore}{\kern0pt}tac\ x\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}rsa{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{in}\ meta{\isacharunderscore}{\kern0pt}spec{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ threelistsappend{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ real{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}a\ {\isasymin}\ set\ rsa{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}drule\ somewhereInside{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ exE{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}\ AALTs\ bs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}\ {\isacharat}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ a\ {\isacharhash}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ rs{\isadigit{2}}\ {\isacharat}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ a\ {\isacharhash}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ distinctBy\ rs\ erase\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}insert\ {\isacharparenleft}{\kern0pt}erase\ a{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}erase\ {\isacharbackquote}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}set\ rs{\isadigit{1}}\ {\isasymunion}\ set\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}\ a\ {\isacharhash}{\kern0pt}\ rs{\isadigit{2}}\ {\isacharat}{\kern0pt}\ \ distinctBy\ rs\ erase\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}insert\ {\isacharparenleft}{\kern0pt}erase\ a{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}erase\ {\isacharbackquote}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}set\ rs{\isadigit{1}}\ {\isasymunion}\ set\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ \isacommand{apply}\isamarkupfalse%
-\ force\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ \isacommand{apply}\isamarkupfalse%
-\ force\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}erase\ {\isacharbackquote}{\kern0pt}\ set\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ insert\ {\isacharparenleft}{\kern0pt}erase\ a{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}erase\ {\isacharbackquote}{\kern0pt}\ set\ rsa{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \ \ \isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}erase\ a\ {\isasymin}\ erase\ {\isacharbackquote}{\kern0pt}set\ rsa{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ a\ {\isacharhash}{\kern0pt}\ distinctBy\ rs\ erase\ {\isacharparenleft}{\kern0pt}insert\ {\isacharparenleft}{\kern0pt}erase\ a{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}erase\ {\isacharbackquote}{\kern0pt}\ set\ rsa{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ distinctBy\ rs\ erase\ {\isacharparenleft}{\kern0pt}insert\ {\isacharparenleft}{\kern0pt}erase\ a{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}erase\ {\isacharbackquote}{\kern0pt}\ set\ rsa{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ force\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}\ append{\isacharunderscore}{\kern0pt}Cons\ append{\isacharunderscore}{\kern0pt}assoc\ append{\isacharunderscore}{\kern0pt}self{\isacharunderscore}{\kern0pt}conv{\isadigit{2}}\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ same{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}eq\ somewhereMapInside{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ force%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\ \isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ alts{\isacharunderscore}{\kern0pt}dBrewrites{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}AALTs\ bs\ rs\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}distinctBy\ rs\ erase\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ rs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ alts{\isacharunderscore}{\kern0pt}dBrewrites{\isacharunderscore}{\kern0pt}withFront\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}Nil\ dB{\isacharunderscore}{\kern0pt}single{\isacharunderscore}{\kern0pt}step\ empty{\isacharunderscore}{\kern0pt}set\ image{\isacharunderscore}{\kern0pt}empty{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\ \ \isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bsimp{\isacharunderscore}{\kern0pt}rewrite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ {\isacharparenleft}{\kern0pt}rrewrites\ r\ {\isacharparenleft}{\kern0pt}\ bsimp\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ r\ rule{\isacharcolon}{\kern0pt}\ bsimp{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bsimp\ r{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ AZERO{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ continuous{\isacharunderscore}{\kern0pt}rewrite\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}{\isasymexists}bs{\isachardot}{\kern0pt}\ bsimp\ r{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ AONE\ bs{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}erule\ exE{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ bsimp{\isacharunderscore}{\kern0pt}ASEQ{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ real{\isacharunderscore}{\kern0pt}trans\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ star{\isacharunderscore}{\kern0pt}seq\ star{\isacharunderscore}{\kern0pt}seq{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}ASEQ{\isadigit{0}}\ bsimp{\isacharunderscore}{\kern0pt}ASEQ{\isadigit{1}}\ real{\isacharunderscore}{\kern0pt}trans\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ rs{\isadigit{2}}\ star{\isacharunderscore}{\kern0pt}seq\ star{\isacharunderscore}{\kern0pt}seq{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{defer}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bsimp{\isacharunderscore}{\kern0pt}aalts{\isacharunderscore}{\kern0pt}simpcases{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ auto\isanewline
-\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}AALTs\ bs{\isadigit{1}}\ rs\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ AALTs\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}map\ bsimp\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}AALTs\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}map\ bsimp\ rs{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ AALTs\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}flts\ {\isacharparenleft}{\kern0pt}map\ bsimp\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}AALTs\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}flts\ {\isacharparenleft}{\kern0pt}map\ bsimp\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ AALTs\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}distinctBy\ {\isacharparenleft}{\kern0pt}flts\ {\isacharparenleft}{\kern0pt}map\ bsimp\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ erase\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}AALTs\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}distinctBy\ {\isacharparenleft}{\kern0pt}flts\ {\isacharparenleft}{\kern0pt}map\ bsimp\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ erase\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}AALTs\ bs{\isadigit{1}}\ {\isacharparenleft}{\kern0pt}distinctBy\ {\isacharparenleft}{\kern0pt}flts\ {\isacharparenleft}{\kern0pt}map\ bsimp\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ erase\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ real{\isacharunderscore}{\kern0pt}trans{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ bsimp{\isacharunderscore}{\kern0pt}AALTsrewrites{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ alts{\isacharunderscore}{\kern0pt}dBrewrites{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ fltsrewrites\ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ alts{\isacharunderscore}{\kern0pt}simpalts\ \isacommand{by}\isamarkupfalse%
-\ force%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rewritenullable{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}r{\isadigit{1}}\ {\isasymleadsto}\ r{\isadigit{2}}{\isacharsemicolon}{\kern0pt}\ bnullable\ r{\isadigit{1}}\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ bnullable\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ r{\isadigit{1}}\ r{\isadigit{2}}\ rule{\isacharcolon}{\kern0pt}\ rrewrite{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{4}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ UnCI\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse\ imageI{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}z{\isadigit{3}}{\isacharparenright}{\kern0pt}\ Un{\isacharunderscore}{\kern0pt}iff\ bnullable{\isacharunderscore}{\kern0pt}correctness\ insert{\isacharunderscore}{\kern0pt}iff\ list{\isachardot}{\kern0pt}set{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ qq{\isadigit{3}}\ set{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rewrite{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}nullable{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}r{\isadigit{1}}\ {\isasymleadsto}\ r{\isadigit{2}}{\isacharsemicolon}{\kern0pt}\ {\isasymnot}bnullable\ r{\isadigit{1}}\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isasymnot}bnullable\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ r{\isadigit{1}}\ r{\isadigit{2}}\ rule{\isacharcolon}{\kern0pt}\ rrewrite{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto\ \isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}{\isacharplus}{\kern0pt}\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rewritesnullable{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}\ r{\isadigit{1}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{2}}{\isacharsemicolon}{\kern0pt}\ bnullable\ r{\isadigit{1}}\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ bnullable\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ r{\isadigit{1}}\ r{\isadigit{2}}\ rule{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ rewritenullable{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nonbnullable{\isacharunderscore}{\kern0pt}lists{\isacharunderscore}{\kern0pt}concat{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ {\isasymlbrakk}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}{\isasymexists}r{\isadigit{0}}{\isasymin}set\ rs{\isadigit{1}}{\isachardot}{\kern0pt}\ bnullable\ r{\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\ {\isasymnot}\ bnullable\ r{\isacharsemicolon}{\kern0pt}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}{\isasymexists}r{\isadigit{0}}{\isasymin}set\ rs{\isadigit{2}}{\isachardot}{\kern0pt}\ bnullable\ r{\isadigit{0}}{\isacharparenright}{\kern0pt}{\isasymrbrakk}\ {\isasymLongrightarrow}\ \isanewline
-{\isasymnot}{\isacharparenleft}{\kern0pt}{\isasymexists}r{\isadigit{0}}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}set\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isacharat}{\kern0pt}rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ bnullable\ r{\isadigit{0}}\ {\isacharparenright}{\kern0pt}\ {\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ nomember{\isacharunderscore}{\kern0pt}bnullable{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}{\isasymexists}r{\isadigit{0}}{\isasymin}set\ rs{\isadigit{1}}{\isachardot}{\kern0pt}\ bnullable\ r{\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\ {\isasymnot}\ bnullable\ r{\isacharsemicolon}{\kern0pt}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}{\isasymexists}r{\isadigit{0}}{\isasymin}set\ rs{\isadigit{2}}{\isachardot}{\kern0pt}\ bnullable\ r{\isadigit{0}}{\isacharparenright}{\kern0pt}{\isasymrbrakk}\isanewline
-\ {\isasymLongrightarrow}\ {\isasymnot}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ nonbnullable{\isacharunderscore}{\kern0pt}lists{\isacharunderscore}{\kern0pt}concat\ qq{\isadigit{3}}\ \isacommand{by}\isamarkupfalse%
-\ presburger%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bnullable{\isacharunderscore}{\kern0pt}segment{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isacharat}{\kern0pt}rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\ bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymor}\ bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymor}\ bnullable\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}{\isasymexists}r{\isadigit{0}}{\isasymin}set\ rs{\isadigit{1}}{\isachardot}{\kern0pt}\ \ bnullable\ r{\isadigit{0}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ qq{\isadigit{3}}\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ r{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}{\isasymexists}r{\isadigit{0}}{\isasymin}set\ rs{\isadigit{2}}{\isachardot}{\kern0pt}\ \ bnullable\ r{\isadigit{0}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ \isacommand{apply}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}False{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ nomember{\isacharunderscore}{\kern0pt}bnullable\ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\ \ \isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ bnullablewhichbmkeps{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}bnullable\ \ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isacharat}{\kern0pt}rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\ {\isasymnot}\ bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\ bnullable\ r\ {\isasymrbrakk}\isanewline
-\ {\isasymLongrightarrow}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isacharat}{\kern0pt}rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bs\ {\isacharat}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bmkeps\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ qq{\isadigit{2}}\ bnullable{\isacharunderscore}{\kern0pt}Hdbmkeps{\isacharunderscore}{\kern0pt}Hd\ \isacommand{by}\isamarkupfalse%
-\ force%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rrewrite{\isacharunderscore}{\kern0pt}nbnullable{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}\ r{\isadigit{1}}\ {\isasymleadsto}\ r{\isadigit{2}}\ {\isacharsemicolon}{\kern0pt}\ {\isasymnot}\ bnullable\ r{\isadigit{1}}\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isasymnot}bnullable\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ rule{\isacharcolon}{\kern0pt}\ rrewrite{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ rewrite{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}nullable\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ spillbmkepslistr{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{1}}\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ {\isasymLongrightarrow}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{1}}\ rs{\isadigit{1}}\ {\isacharhash}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ bnullable{\isacharunderscore}{\kern0pt}Hdbmkeps{\isacharunderscore}{\kern0pt}Hd{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bmkeps{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ k{\isadigit{0}}a\ list{\isachardot}{\kern0pt}set{\isacharunderscore}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ qq{\isadigit{1}}\ qq{\isadigit{4}}\ qs{\isadigit{3}}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ third{\isacharunderscore}{\kern0pt}segment{\isacharunderscore}{\kern0pt}bnullable{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}rs{\isadigit{2}}{\isacharat}{\kern0pt}rs{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\ {\isasymnot}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\ {\isasymnot}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isasymrbrakk}\ {\isasymLongrightarrow}\ \isanewline
-bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ \isanewline
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isachardot}{\kern0pt}left{\isacharunderscore}{\kern0pt}neutral\ append{\isacharunderscore}{\kern0pt}Cons\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}segment\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{7}}{\isacharparenright}{\kern0pt}\ rrewrite{\isacharunderscore}{\kern0pt}nbnullable{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ third{\isacharunderscore}{\kern0pt}segment{\isacharunderscore}{\kern0pt}bmkeps{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}{\isasymlbrakk}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}rs{\isadigit{2}}{\isacharat}{\kern0pt}rs{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\ {\isasymnot}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\ {\isasymnot}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isasymrbrakk}\ {\isasymLongrightarrow}\ \isanewline
-bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}rs{\isadigit{2}}{\isacharat}{\kern0pt}rs{\isadigit{3}}{\isacharparenright}{\kern0pt}\ {\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}{\isasymforall}r\ {\isasymin}\ set\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymnot}bnullable\ r{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}rs{\isadigit{2}}{\isacharat}{\kern0pt}rs{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isacharat}{\kern0pt}rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharat}{\kern0pt}rs{\isadigit{3}}{\isacharparenright}{\kern0pt}\ {\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ qq{\isadigit{2}}\ qq{\isadigit{3}}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isachardot}{\kern0pt}assoc{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isachardot}{\kern0pt}assoc\ in{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}conv{\isacharunderscore}{\kern0pt}decomp\ r{\isadigit{2}}\ third{\isacharunderscore}{\kern0pt}segment{\isacharunderscore}{\kern0pt}bnullable{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ third{\isacharunderscore}{\kern0pt}segment{\isacharunderscore}{\kern0pt}bnullable\ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rewrite{\isacharunderscore}{\kern0pt}bmkepsalt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ {\isasymlbrakk}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ AALTs\ bs{\isadigit{1}}\ rs{\isadigit{1}}\ {\isacharhash}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\ bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isasymrbrakk}\isanewline
-\ \ \ \ \ \ \ {\isasymLongrightarrow}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ AALTs\ bs{\isadigit{1}}\ rs{\isadigit{1}}\ {\isacharhash}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rsa{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ qq{\isadigit{1}}\ \isacommand{apply}\isamarkupfalse%
-\ force\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{1}}\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ qq{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ r{\isadigit{2}}\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isanewline
-\ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ list{\isachardot}{\kern0pt}set{\isacharunderscore}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}\ append{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}conv{\isadigit{2}}\ list{\isachardot}{\kern0pt}set{\isacharunderscore}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ qq{\isadigit{2}}\ qq{\isadigit{3}}\ rewritenullable\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{8}}{\isacharparenright}{\kern0pt}\ self{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}conv{\isadigit{2}}\ spillbmkepslistr{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\isanewline
-\ \ \isacommand{thm}\isamarkupfalse%
-\ qq{\isadigit{1}}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bmkeps\ \ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ AALTs\ bs{\isadigit{1}}\ rs{\isadigit{1}}\ {\isacharhash}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bmkeps\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rsb{\isacharparenright}{\kern0pt}\ {\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}Cons\ append{\isacharunderscore}{\kern0pt}Nil\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ bnullable{\isacharunderscore}{\kern0pt}segment\ rewritenullable\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ third{\isacharunderscore}{\kern0pt}segment{\isacharunderscore}{\kern0pt}bmkeps{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ rewrite{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}nullable\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ third{\isacharunderscore}{\kern0pt}segment{\isacharunderscore}{\kern0pt}bmkeps{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rewrite{\isacharunderscore}{\kern0pt}bmkeps{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}\ r{\isadigit{1}}\ {\isasymleadsto}\ r{\isadigit{2}}{\isacharsemicolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bnullable\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isasymrbrakk}\ {\isasymLongrightarrow}\ bmkeps\ r{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ bmkeps\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\isanewline
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}frule\ rewritenullable{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ r{\isadigit{1}}\ r{\isadigit{2}}\ rule{\isacharcolon}{\kern0pt}\ rrewrite{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ b{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}frule\ bnullable{\isacharunderscore}{\kern0pt}segment{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ qq{\isadigit{1}}\ \isacommand{apply}\isamarkupfalse%
-\ force\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ r{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bnullablewhichbmkeps\ rewritenullable\ \isacommand{apply}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}{\isasymnot}\ bnullable\ r{\isacharprime}{\kern0pt}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ qq{\isadigit{2}}\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rrewrite{\isacharunderscore}{\kern0pt}nbnullable\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ flts{\isacharunderscore}{\kern0pt}append\ qs{\isadigit{3}}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ rewrite{\isacharunderscore}{\kern0pt}bmkepsalt{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ q{\isadigit{3}}a\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ q{\isadigit{3}}a{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bnullable{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ b{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}z{\isadigit{3}}{\isacharparenright}{\kern0pt}\ Un{\isacharunderscore}{\kern0pt}iff\ bnullable{\isacharunderscore}{\kern0pt}correctness\ erase{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}\ qq{\isadigit{1}}\ qq{\isadigit{2}}\ qq{\isadigit{3}}\ set{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rewrites{\isacharunderscore}{\kern0pt}bmkeps{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymlbrakk}\ {\isacharparenleft}{\kern0pt}r{\isadigit{1}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bnullable\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isasymrbrakk}\ {\isasymLongrightarrow}\ bmkeps\ r{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ bmkeps\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ r{\isadigit{1}}\ r{\isadigit{2}}\ rule{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ r{\isadigit{2}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}metis\ rewritesnullable{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bmkeps\ r{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ bmkeps\ r{\isadigit{2}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rewrite{\isacharunderscore}{\kern0pt}bmkeps\ \isacommand{by}\isamarkupfalse%
-\ presburger%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{thm}\isamarkupfalse%
-\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ alts{\isacharunderscore}{\kern0pt}rewrite{\isacharunderscore}{\kern0pt}front{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}r\ {\isasymleadsto}\ r{\isacharprime}{\kern0pt}\ {\isasymLongrightarrow}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}r\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}r{\isacharprime}{\kern0pt}\ {\isacharhash}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}Cons\ append{\isacharunderscore}{\kern0pt}Nil\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ alt{\isacharunderscore}{\kern0pt}rewrite{\isacharunderscore}{\kern0pt}front{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}r\ {\isasymleadsto}\ r{\isacharprime}{\kern0pt}\ {\isasymLongrightarrow}\ AALT\ bs\ r\ r{\isadigit{2}}\ {\isasymleadsto}\ AALT\ bs\ r{\isacharprime}{\kern0pt}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ alts{\isacharunderscore}{\kern0pt}rewrite{\isacharunderscore}{\kern0pt}front\ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ to{\isacharunderscore}{\kern0pt}zero{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ AALT\ bs\ {\isacharparenleft}{\kern0pt}ASEQ\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ AZERO\ r{\isacharparenright}{\kern0pt}\ r{\isadigit{2}}\ {\isasymleadsto}\ \ AALT\ bs\ AZERO\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ alts{\isacharunderscore}{\kern0pt}rewrite{\isacharunderscore}{\kern0pt}front\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ alt{\isacharunderscore}{\kern0pt}remove{\isadigit{0}}{\isacharunderscore}{\kern0pt}front{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ AALT\ bs\ AZERO\ r\ {\isasymleadsto}\ AALTs\ bs\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ rrewrite{\isadigit{0}}away{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ alt{\isacharunderscore}{\kern0pt}rewrites{\isacharunderscore}{\kern0pt}back{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}r{\isadigit{2}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{2}}{\isacharprime}{\kern0pt}\ {\isasymLongrightarrow}AALT\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ AALT\ bs\ r{\isadigit{1}}\ r{\isadigit{2}}{\isacharprime}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ r{\isadigit{2}}\ r{\isadigit{2}}{\isacharprime}{\kern0pt}\ arbitrary{\isacharcolon}{\kern0pt}\ bs\ rule{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ rs{\isadigit{1}}\ rs{\isadigit{2}}\ srewrites{\isacharunderscore}{\kern0pt}alt{\isadigit{1}}\ ss{\isadigit{1}}\ ss{\isadigit{2}}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rewrite{\isacharunderscore}{\kern0pt}fuse{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ r{\isadigit{2}}\ {\isasymleadsto}\ r{\isadigit{3}}\ {\isasymLongrightarrow}\ fuse\ bs\ r{\isadigit{2}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ fuse\ bs\ r{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ r{\isadigit{2}}\ r{\isadigit{3}}\ arbitrary{\isacharcolon}{\kern0pt}\ bs\ rule{\isacharcolon}{\kern0pt}\ rrewrite{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ auto\isanewline
-\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ continuous{\isacharunderscore}{\kern0pt}rewrite{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ fuse{\isacharunderscore}{\kern0pt}append\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ star{\isacharunderscore}{\kern0pt}seq\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ star{\isacharunderscore}{\kern0pt}seq{\isadigit{2}}\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ contextrewrites{\isadigit{2}}\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{7}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{8}}{\isacharparenright}{\kern0pt}\ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}assoc\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{9}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ append{\isacharunderscore}{\kern0pt}assoc\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ fuse{\isacharunderscore}{\kern0pt}append\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\ \ \isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rewrites{\isacharunderscore}{\kern0pt}fuse{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}r{\isadigit{2}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{2}}{\isacharprime}{\kern0pt}\ {\isasymLongrightarrow}\ \ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{1}}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ \ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{1}}\ r{\isadigit{2}}{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ r{\isadigit{2}}\ r{\isadigit{2}}{\isacharprime}{\kern0pt}\ arbitrary{\isacharcolon}{\kern0pt}\ bs{\isadigit{1}}\ rule{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ real{\isacharunderscore}{\kern0pt}trans\ rewrite{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ \ bder{\isacharunderscore}{\kern0pt}fuse{\isacharunderscore}{\kern0pt}list{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ map\ {\isacharparenleft}{\kern0pt}bder\ c\ {\isasymcirc}\ fuse\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{1}}\ {\isasymcirc}\ bder\ c{\isacharparenright}{\kern0pt}\ rs{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bder{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rewrite{\isacharunderscore}{\kern0pt}der{\isacharunderscore}{\kern0pt}altmiddle{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}bder\ c\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ AALTs\ bs{\isadigit{1}}\ rs{\isadigit{1}}\ {\isacharhash}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ bder\ c\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isadigit{1}}\ {\isacharat}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ bder{\isacharunderscore}{\kern0pt}fuse{\isacharunderscore}{\kern0pt}list{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ many{\isacharunderscore}{\kern0pt}steps{\isacharunderscore}{\kern0pt}later{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{8}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lock{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}der{\isacharunderscore}{\kern0pt}removal{\isacharcolon}{\kern0pt}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}\ erase\ a{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ erase\ a{\isadigit{2}}\ {\isasymLongrightarrow}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ bder\ c\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}a{\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rsb\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}a{\isadigit{2}}{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rsc{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ bder\ c\ {\isacharparenleft}{\kern0pt}AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}a{\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rsb\ {\isacharat}{\kern0pt}\ rsc{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rewrite{\isacharunderscore}{\kern0pt}after{\isacharunderscore}{\kern0pt}der{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}r{\isadigit{1}}\ {\isasymleadsto}\ r{\isadigit{2}}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ r{\isadigit{1}}\ r{\isadigit{2}}\ arbitrary{\isacharcolon}{\kern0pt}\ c\ rule{\isacharcolon}{\kern0pt}\ rrewrite{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ contextrewrites{\isadigit{1}}\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ rrewrite{\isadigit{0}}away\ rs{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ many{\isacharunderscore}{\kern0pt}steps{\isacharunderscore}{\kern0pt}later{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ to{\isacharunderscore}{\kern0pt}zero{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alt{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ many{\isacharunderscore}{\kern0pt}steps{\isacharunderscore}{\kern0pt}later{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ alt{\isacharunderscore}{\kern0pt}remove{\isadigit{0}}{\isacharunderscore}{\kern0pt}front{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ many{\isacharunderscore}{\kern0pt}steps{\isacharunderscore}{\kern0pt}later{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ bder{\isacharunderscore}{\kern0pt}fuse\ fuse{\isacharunderscore}{\kern0pt}append\ rs{\isadigit{1}}\ \isacommand{apply}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ r{\isadigit{1}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}{\isasymnot}bnullable\ r{\isadigit{2}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rewrite{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}nullable\ \isacommand{apply}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp{\isacharplus}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ star{\isacharunderscore}{\kern0pt}seq\ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ r{\isadigit{2}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp{\isacharplus}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bmkeps\ r{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ bmkeps\ r{\isadigit{2}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rewrite{\isacharunderscore}{\kern0pt}bmkeps\ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ contextrewrites{\isadigit{1}}\ star{\isacharunderscore}{\kern0pt}seq\ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rewritenullable\ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}case{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}bnullable\ r{\isadigit{1}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subgoal{\isacharunderscore}{\kern0pt}tac\ {\isachardoublequoteopen}ASEQ\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ r{\isadigit{3}}\ {\isasymleadsto}\ ASEQ\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ r{\isadigit{4}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{prefer}\isamarkupfalse%
-\ {\isadigit{2}}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}\ \isacommand{apply}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ many{\isacharunderscore}{\kern0pt}steps{\isacharunderscore}{\kern0pt}later{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ alt{\isacharunderscore}{\kern0pt}rewrite{\isacharunderscore}{\kern0pt}front{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ assumption\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ alt{\isacharunderscore}{\kern0pt}rewrites{\isacharunderscore}{\kern0pt}back\ rewrites{\isacharunderscore}{\kern0pt}fuse{\isacharparenright}{\kern0pt}\ \isanewline
-\isanewline
-\ \ \ \ \ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ contextrewrites{\isadigit{2}}\ \isacommand{apply}\isamarkupfalse%
-\ force\isanewline
-\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{7}}{\isacharparenright}{\kern0pt}\ \isacommand{apply}\isamarkupfalse%
-\ force\isanewline
-\ \ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ rewrite{\isacharunderscore}{\kern0pt}der{\isacharunderscore}{\kern0pt}altmiddle\ \isacommand{apply}\isamarkupfalse%
-\ auto{\isacharbrackleft}{\kern0pt}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bder{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ bder{\isacharunderscore}{\kern0pt}fuse{\isacharunderscore}{\kern0pt}list\ map{\isacharunderscore}{\kern0pt}map\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{9}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ List{\isachardot}{\kern0pt}map{\isachardot}{\kern0pt}compositionality\ bder{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ bder{\isacharunderscore}{\kern0pt}fuse{\isacharunderscore}{\kern0pt}list\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{0}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ r{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}rstar\ rrewrite{\isachardot}{\kern0pt}intros{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bder{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ bder{\isacharunderscore}{\kern0pt}bsimp{\isacharunderscore}{\kern0pt}AALTs\ bsimp{\isacharunderscore}{\kern0pt}AALTs{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}AALTsrewrites{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ lock{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}der{\isacharunderscore}{\kern0pt}removal\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rewrites{\isacharunderscore}{\kern0pt}after{\isacharunderscore}{\kern0pt}der{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ \ r{\isadigit{1}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{2}}\ \ {\isasymLongrightarrow}\ \ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bder\ c\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induction\ r{\isadigit{1}}\ r{\isadigit{2}}\ rule{\isacharcolon}{\kern0pt}\ rrewrites{\isachardot}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}rule\ rs{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ real{\isacharunderscore}{\kern0pt}trans\ rewrite{\isacharunderscore}{\kern0pt}after{\isacharunderscore}{\kern0pt}der{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\ \ \isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ central{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ {\isacharparenleft}{\kern0pt}bders\ r\ s{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ \ {\isacharparenleft}{\kern0pt}bders{\isacharunderscore}{\kern0pt}simp\ r\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}induct\ s\ arbitrary{\isacharcolon}{\kern0pt}\ r\ rule{\isacharcolon}{\kern0pt}\ rev{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ bders{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}subst\ bders{\isacharunderscore}{\kern0pt}simp{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ bders{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ bders{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ bders{\isacharunderscore}{\kern0pt}simp{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ bders{\isacharunderscore}{\kern0pt}simp{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ bsimp{\isacharunderscore}{\kern0pt}rewrite\ real{\isacharunderscore}{\kern0pt}trans\ rewrites{\isacharunderscore}{\kern0pt}after{\isacharunderscore}{\kern0pt}der{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{thm}\isamarkupfalse%
-\ arexp{\isachardot}{\kern0pt}induct\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ quasi{\isacharunderscore}{\kern0pt}main{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}bnullable\ {\isacharparenleft}{\kern0pt}bders\ r\ s{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\ bmkeps\ {\isacharparenleft}{\kern0pt}bders\ r\ s{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ bmkeps\ {\isacharparenleft}{\kern0pt}bders{\isacharunderscore}{\kern0pt}simp\ r\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ central\ rewrites{\isacharunderscore}{\kern0pt}bmkeps\ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ main{\isacharunderscore}{\kern0pt}main{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blexer\ r\ s\ {\isacharequal}{\kern0pt}\ blexer{\isacharunderscore}{\kern0pt}simp\ r\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ b{\isadigit{4}}\ blexer{\isacharunderscore}{\kern0pt}def\ blexer{\isacharunderscore}{\kern0pt}simp{\isacharunderscore}{\kern0pt}def\ quasi{\isacharunderscore}{\kern0pt}main{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ blexersimp{\isacharunderscore}{\kern0pt}correctness{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}blexer{\isacharunderscore}{\kern0pt}simp\ r\ s{\isacharequal}{\kern0pt}\ lexer\ r\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ blexer{\isacharunderscore}{\kern0pt}correctness\ main{\isacharunderscore}{\kern0pt}main\ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{unused{\isacharunderscore}{\kern0pt}thms}\isamarkupfalse%
-\isanewline
-\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
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