50 \isamarkupsection{Introduction%

    50 \isamarkupsection{Core of the proof%

    72 \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}}

    72 \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}}

    75 This lemma links the derivative of a bit-coded regular expression with

    75 This lemma establishes a link with the lexer without bit-codes.

    76 the regular expression itself before the derivative. 

    76 

    77 

    77 With it we get the correctness of bit-coded algorithm.

    78 \begin{lemma}

    79 \isa{lexer\mbox{$_b$}\ r\ s\ {\isacharequal}{\kern0pt}\ lexer\ r\ s}

    80 \end{lemma}

    81 

    82 However what is not certain is whether we can add simplification

    83 to the bit-coded algorithm, without breaking the correct lexing output.

    84 This might sound trivial in the case of producing a YES/NO answer,

    85 but once we require a lexing output to be produced (which is required

    86 in applications like compiler front-end, malicious attack domain extraction, 

    87 etc.), it is not straightforward if we still extract what is needed according

    88 to the POSIX standard.

    89 

    90 By simplification, we mean specifically the following rules:

    91 

    92 \begin{center}

    93   \begin{tabular}{lcl}

    94   \isa{\mbox{}\inferrule{\mbox{}}{\mbox{ASEQ\ bs\ AZERO\ r\isactrlsub {\isadigit{2}}\ {\isasymleadsto}\ AZERO}}}\\

    95   \isa{\mbox{}\inferrule{\mbox{}}{\mbox{ASEQ\ bs\ r\isactrlsub {\isadigit{1}}\ AZERO\ {\isasymleadsto}\ AZERO}}}\\

    96   \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}}}}}\\

    97   \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}}}}}\\

    98   \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}}}}}\\

    99   \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}}}}\\

   100   \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}}}}\\

   101   \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}}}}\\

   102   \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}}}\\

   103   \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}}}}\\

   104   \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymleadsto}\ AZERO}}}\\

   105   \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharbrackleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isasymleadsto}\ fuse\ bs\ r\isactrlsub {\isadigit{1}}}}}\\

   106   \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}}}}\\

   107 

   108 

   109   \isa{\mbox{}\inferrule{\mbox{}}{\mbox{Empty\ {\isacharcolon}{\kern0pt}\ \isactrlbold {\isadigit{1}}}}} & 

   110   \isa{\mbox{}\inferrule{\mbox{}}{\mbox{Char\ c\ {\isacharcolon}{\kern0pt}\ c}}}\\[4mm]

   111   \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}}}}} &

   112   \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]

   113   \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}}}}}  &

   114   \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}}}}

   115 

   116 

   117   \end{tabular}

   118 \end{center}

   119 

   120 

   121 And these can be made compact by the following simplification function:

   122 

   123 \begin{center}

   124   \begin{tabular}{lcl}

   125   \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}}\\

   126   \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}}\\

   127   \isa{bsimp\ AZERO} & $\dn$ & \isa{AZERO}\\

   128   \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}}\\

   129 \end{tabular}

   130 \end{center}

   131 

   132 The core idea of the proof is that two regular expressions,

   133 if "isomorphic" up to a finite number of rewrite steps, will

   134 remain so when we take derivative on both of them.

   135 This can be expressed by the following rewrite relation lemma:

   136 \begin{lemma}

   137 \isa{{\isacharparenleft}{\kern0pt}r\mbox{$\bbslash$}s{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ bders{\isacharunderscore}{\kern0pt}simp\ r\ s}

   138 \end{lemma}%

   139 \end{isamarkuptext}\isamarkuptrue%

   140 %

   141 \isadelimdocument

   142 %

   143 \endisadelimdocument

   144 %

   145 \isatagdocument

   146 %

   147 \isamarkupsection{Introduction%

   148 }

   149 \isamarkuptrue%

   150 %

   151 \endisatagdocument

   152 {\isafolddocument}%

   153 %

   154 \isadelimdocument

   155 %

   156 \endisadelimdocument

   157 %

   158 \begin{isamarkuptext}%

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  1826 %:%234=300%:%

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  1761 %:%231=312%:%

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  1763 %:%233=314%:%

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  1766 %:%236=317%:%

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  1767 %:%237=318%:%

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  1769 %:%239=320%:%

  1850 %:%258=324%:%

  1770 %:%240=321%:%

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  1771 %:%241=322%:%

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  1772 %:%242=323%:%

  1853 %:%261=327%:%

  1773 %:%243=324%:%

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  1774 %:%244=325%:%

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  1857 %:%265=331%:%

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  1778 %:%248=329%:%

  1859 %:%267=333%:%

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  1860 %:%268=334%:%

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  1870 %:%278=344%:%

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  1880 %:%288=354%:%

  1800 %:%288=355%:%

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  1810 %:%298=365%:%

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  1812 %:%300=367%:%

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  1900 %:%308=374%:%

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  1828 %:%315=383%:%

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  1910 %:%318=384%:%

  1830 %:%316=385%:%

  1911 %:%319=385%:%

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  1912 %:%320=386%:%

  1832 %:%318=387%:%

  1913 %:%321=387%:%

  1833 %:%319=388%:%

  1914 %:%322=388%:%

  1834 %:%320=389%:%

  1915 %:%323=389%:%

  1835 %:%321=390%:%

  1916 %:%324=390%:%

  1836 %:%322=391%:%

  1917 %:%325=391%:%

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  1918 %:%326=392%:%

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  1919 %:%327=393%:%

  1839 %:%325=394%:%

  1920 %:%328=394%:%

  1840 %:%326=395%:%

  1921 %:%329=395%:%

  1841 %:%327=396%:%

  1922 %:%330=396%:%

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  1923 %:%331=397%:%

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  1924 %:%340=404%:%

  1844 %:%330=399%:%

  1925 %:%352=406%:%

  1845 %:%331=400%:%

  1926 %:%353=407%:%

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  1927 %:%353=408%:%

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  1929 %:%355=410%:%

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  1930 %:%356=411%:%

  1850 %:%336=405%:%

  1931 %:%357=412%:%

  1851 %:%337=406%:%

  1932 %:%358=413%:%

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  1933 %:%359=414%:%

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  1934 %:%360=415%:%

  1854 %:%340=409%:%

  1935 %:%361=416%:%

  1855 %:%341=410%:%

  1936 %:%362=417%:%

  1856 %:%342=411%:%

  1937 %:%363=418%:%

  1857 %:%343=412%:%

  1938 %:%364=419%:%

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  1939 %:%365=420%:%

  1859 %:%345=414%:%

  1940 %:%366=421%:%

  1860 %:%346=415%:%

  1941 %:%367=422%:%

  1861 %:%347=416%:%

  1942 %:%368=423%:%

  1862 %:%348=417%:%

  1943 %:%369=424%:%

  1863 %:%349=418%:%

  1944 %:%370=425%:%

  1864 %:%350=419%:%

  1945 %:%371=426%:%

  1865 %:%351=420%:%

  1946 %:%372=427%:%

  1866 %:%352=421%:%

  1947 %:%373=428%:%

  1867 %:%353=422%:%

  1948 %:%374=429%:%

  1868 %:%354=423%:%

  1949 %:%375=430%:%

  1869 %:%355=424%:%

  1950 %:%376=431%:%

  1870 %:%356=425%:%

  1951 %:%377=432%:%

  1871 %:%357=426%:%

  1952 %:%378=433%:%

  1872 %:%358=427%:%

  1953 %:%379=434%:%

  1873 %:%359=428%:%

  1954 %:%380=435%:%

  1874 %:%360=429%:%

  1955 %:%381=436%:%

  1875 %:%361=430%:%

  1956 %:%382=437%:%

  1876 %:%362=431%:%

  1957 %:%383=438%:%

  1877 %:%363=432%:%

  1958 %:%384=439%:%

  1878 %:%364=433%:%

  1959 %:%385=440%:%

  1879 %:%365=434%:%

  1960 %:%386=441%:%

  1880 %:%366=435%:%

  1961 %:%387=442%:%

  1881 %:%367=436%:%

  1962 %:%388=443%:%

  1882 %:%368=437%:%

  1963 %:%389=444%:%

  1883 %:%369=438%:%

  1964 %:%390=445%:%

  1884 %:%370=439%:%

  1965 %:%391=446%:%

  1885 %:%371=440%:%

  1966 %:%392=447%:%

  1886 %:%372=441%:%

  1967 %:%392=448%:%

  1887 %:%373=442%:%

  1968 %:%393=449%:%

  1888 %:%374=443%:%

  1969 %:%394=450%:%

  1889 %:%375=444%:%

  1970 %:%395=451%:%

  1890 %:%376=445%:%

  1971 %:%396=452%:%

  1891 %:%377=446%:%

  1972 %:%397=453%:%

  1892 %:%378=447%:%

  1973 %:%397=454%:%

  1893 %:%379=448%:%

  1974 %:%398=455%:%

  1894 %:%380=449%:%

  1975 %:%399=456%:%

  1895 %:%381=450%:%

  1976 %:%400=457%:%

  1896 %:%382=451%:%

  1977 %:%401=458%:%

  1897 %:%383=452%:%

  1978 %:%402=459%:%

  1898 %:%384=453%:%

  1979 %:%403=460%:%

  1899 %:%385=454%:%

  1980 %:%404=461%:%

  1900 %:%386=455%:%

  1981 %:%405=462%:%

  1901 %:%387=456%:%

  1982 %:%406=463%:%

  1902 %:%388=457%:%

  1983 %:%407=464%:%

  1903 %:%389=458%:%

  1984 %:%408=465%:%

  1904 %:%390=459%:%

  1985 %:%409=466%:%

  1905 %:%391=460%:%

  1986 %:%410=467%:%

  1906 %:%392=461%:%

  1987 %:%411=468%:%

  1907 %:%393=462%:%

  1988 %:%412=469%:%

  1908 %:%394=463%:%

  1989 %:%413=470%:%

  1909 %:%395=464%:%

  1990 %:%414=471%:%

  1910 %:%396=465%:%

  1991 %:%415=472%:%

  1911 %:%397=466%:%

  1992 %:%416=473%:%

  1912 %:%398=467%:%

  1993 %:%417=474%:%

  1913 %:%399=468%:%

  1994 %:%418=475%:%

  1914 %:%400=469%:%

  1995 %:%419=476%:%

  1915 %:%401=470%:%

  1996 %:%420=477%:%

  1916 %:%402=471%:%

  1997 %:%421=478%:%

  1917 %:%403=472%:%

  1998 %:%422=479%:%

  1918 %:%404=473%:%

  1999 %:%423=480%:%

  1919 %:%405=474%:%

  2000 %:%424=481%:%

  1920 %:%414=478%:%

  2001 %:%425=482%:%

  1921 %:%426=482%:%

  2002 %:%426=483%:%

  1922 %:%427=483%:%

  2003 %:%427=484%:%

  1923 %:%428=484%:%

  2004 %:%428=485%:%

  1924 %:%429=485%:%

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  1925 %:%430=486%:%

  2006 %:%430=487%:%

  1926 %:%431=487%:%

  2007 %:%431=488%:%

  1927 %:%432=488%:%

  2008 %:%432=489%:%

  1928 %:%433=489%:%

  2009 %:%433=490%:%

  1929 %:%434=490%:%

  2010 %:%434=491%:%

  1930 %:%435=491%:%

  2011 %:%435=492%:%

  1931 %:%436=492%:%

  2012 %:%436=493%:%

  1932 %:%437=493%:%

  2013 %:%437=494%:%

  1933 %:%438=494%:%

  2014 %:%438=495%:%

  1934 %:%439=495%:%

  2015 %:%439=496%:%

  1935 %:%440=496%:%

  2016 %:%440=497%:%

  1936 %:%441=497%:%

  2017 %:%441=498%:%

  1937 %:%442=498%:%

  2018 %:%442=499%:%

  1938 %:%443=499%:%

  2019 %:%443=500%:%

  1939 %:%444=500%:%

  2020 %:%444=501%:%

  1940 %:%445=501%:%

  2021 %:%445=502%:%

  1941 %:%446=502%:%

  2022 %:%446=503%:%

  1942 %:%447=503%:%

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  1943 %:%448=504%:%

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  1944 %:%449=505%:%

  2025 %:%449=506%:%

  1945 %:%450=506%:%

  2026 %:%450=507%:%

  1946 %:%451=507%:%

  2027 %:%451=508%:%

  1947 %:%452=508%:%

  2028 %:%452=509%:%

  1948 %:%453=509%:%

  2029 %:%453=510%:%

  1949 %:%454=510%:%

  2030 %:%454=511%:%

  1950 %:%455=511%:%

  2031 %:%455=512%:%

  1951 %:%456=512%:%

  2032 %:%456=513%:%

  1952 %:%457=513%:%

  2033 %:%457=514%:%

  1953 %:%458=514%:%

  2034 %:%458=515%:%

  1954 %:%459=515%:%

  2035 %:%459=516%:%

  1955 %:%460=516%:%

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  1956 %:%461=517%:%

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  1957 %:%462=518%:%

  2038 %:%462=519%:%

  1958 %:%463=519%:%

  2039 %:%463=520%:%

  1959 %:%464=520%:%

  2040 %:%464=521%:%

  1960 %:%465=521%:%

  2041 %:%465=522%:%

  1961 %:%466=522%:%

  2042 %:%466=523%:%

  1962 %:%467=523%:%

  2043 %:%467=524%:%

  1963 %:%468=524%:%

  2044 %:%468=525%:%

  1964 %:%469=525%:%

  2045 %:%469=526%:%

  1965 %:%470=526%:%

  2046 %:%470=527%:%

  1966 %:%471=527%:%

  2047 %:%471=528%:%

  1967 %:%472=528%:%

  2048 %:%472=529%:%

  1968 %:%473=529%:%

  2049 %:%473=530%:%

  1969 %:%474=530%:%

  2050 %:%474=531%:%

  1970 %:%475=531%:%

  2051 %:%475=532%:%

  1971 %:%476=532%:%

  2052 %:%476=533%:%

  1972 %:%477=533%:%

  2053 %:%477=534%:%

  1973 %:%478=534%:%

  2054 %:%478=535%:%

  1974 %:%479=535%:%

  2055 %:%479=536%:%

  1975 %:%480=536%:%

  2056 %:%480=537%:%

  1976 %:%481=537%:%

  2057 %:%481=538%:%

  1977 %:%482=538%:%

  2058 %:%482=539%:%

  1978 %:%483=539%:%

  2059 %:%483=540%:%

  1979 %:%484=540%:%

  2060 %:%484=541%:%

  1980 %:%485=541%:%

  2061 %:%485=542%:%

  1981 %:%486=542%:%

  2062 %:%486=543%:%

  1982 %:%487=543%:%

  2063 %:%495=547%:%

  1983 %:%488=544%:%

  2064 %:%507=551%:%

  1984 %:%489=545%:%

  2065 %:%508=552%:%

  1985 %:%490=546%:%

  2066 %:%509=553%:%

  1986 %:%491=547%:%

  2067 %:%510=554%:%

  1987 %:%492=548%:%

  2068 %:%511=555%:%

  1988 %:%493=549%:%

  2069 %:%512=556%:%

  1989 %:%494=550%:%

  2070 %:%513=557%:%

  1990 %:%495=551%:%

  2071 %:%514=558%:%

  1991 %:%496=552%:%

  2072 %:%515=559%:%

  1992 %:%497=553%:%

  2073 %:%516=560%:%

  1993 %:%498=554%:%

  2074 %:%517=561%:%

  1994 %:%499=555%:%

  2075 %:%518=562%:%

  1995 %:%500=556%:%

  2076 %:%519=563%:%

  1996 %:%501=557%:%

  2077 %:%520=564%:%

  1997 %:%502=558%:%

  2078 %:%521=565%:%

  1998 %:%503=559%:%

  2079 %:%522=566%:%

  1999 %:%504=560%:%

  2080 %:%523=567%:%

  2000 %:%505=561%:%

  2081 %:%524=568%:%

  2001 %:%506=562%:%

  2082 %:%525=569%:%

  2002 %:%507=563%:%

  2083 %:%526=570%:%

  2003 %:%508=564%:%

  2084 %:%527=571%:%

  2004 %:%509=565%:%

  2085 %:%528=572%:%

  2005 %:%510=566%:%

  2086 %:%529=573%:%

  2006 %:%511=567%:%

  2087 %:%530=574%:%

  2007 %:%512=568%:%

  2088 %:%531=575%:%

  2008 %:%513=569%:%

  2089 %:%532=576%:%

  2009 %:%514=570%:%

  2090 %:%533=577%:%

  2010 %:%515=571%:%

  2091 %:%534=578%:%

  2011 %:%516=572%:%

  2092 %:%535=579%:%

  2012 %:%517=573%:%

  2093 %:%536=580%:%

  2013 %:%518=574%:%

  2094 %:%537=581%:%

  2014 %:%519=575%:%

  2095 %:%538=582%:%

  2015 %:%520=576%:%

  2096 %:%539=583%:%

  2016 %:%521=577%:%

  2097 %:%540=584%:%

  2017 %:%522=578%:%

  2098 %:%541=585%:%

  2018 %:%523=579%:%

  2099 %:%542=586%:%

  2019 %:%524=580%:%

  2100 %:%543=587%:%

  2020 %:%525=581%:%

  2101 %:%544=588%:%

  2021 %:%526=582%:%

  2102 %:%545=589%:%

  2022 %:%527=583%:%

  2103 %:%546=590%:%

  2023 %:%528=584%:%

  2104 %:%547=591%:%

  2024 %:%529=585%:%

  2105 %:%548=592%:%

  2025 %:%530=586%:%

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  2026 %:%531=587%:%

  2107 %:%550=594%:%

  2027 %:%532=588%:%

  2108 %:%551=595%:%

  2028 %:%533=589%:%

  2109 %:%552=596%:%

  2029 %:%534=590%:%

  2110 %:%553=597%:%

  2030 %:%535=591%:%

  2111 %:%554=598%:%

  2031 %:%536=592%:%

  2112 %:%555=599%:%

  2032 %:%537=593%:%

  2113 %:%556=600%:%

  2033 %:%538=594%:%

  2114 %:%557=601%:%

  2034 %:%539=595%:%

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  2038 %:%541=599%:%

  2119 %:%562=606%:%

  2039 %:%542=600%:%

  2120 %:%563=607%:%

  2040 %:%543=601%:%

  2121 %:%564=608%:%

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  2122 %:%565=609%:%

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  2123 %:%566=610%:%

  2043 %:%545=604%:%

  2124 %:%567=611%:%

  2044 %:%546=605%:%

  2125 %:%568=612%:%

  2045 %:%547=606%:%

  2126 %:%569=613%:%

  2046 %:%548=607%:%

  2127 %:%570=614%:%

  2047 %:%549=608%:%

  2128 %:%571=615%:%

  2048 %:%550=609%:%

  2129 %:%572=616%:%

  2049 %:%551=610%:%

  2130 %:%573=617%:%

  2050 %:%552=611%:%

  2131 %:%574=618%:%

  2051 %:%553=612%:%

  2132 %:%575=619%:%

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  2133 %:%576=620%:%

  2053 %:%555=614%:%

  2134 %:%577=621%:%

  2054 %:%556=615%:%

  2135 %:%578=622%:%

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  2136 %:%579=623%:%

  2056 %:%558=617%:%

  2137 %:%580=624%:%

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  2138 %:%581=625%:%

  2058 %:%560=619%:%

  2139 %:%582=626%:%

  2059 %:%561=620%:%

  2140 %:%583=627%:%

  2060 %:%562=621%:%

  2141 %:%584=628%:%

  2061 %:%563=622%:%

  2142 %:%585=629%:%

  2062 %:%564=623%:%

  2143 %:%586=630%:%

  2063 %:%565=624%:%

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  2064 %:%566=625%:%

  2145 %:%588=632%:%

  2065 %:%567=626%:%

  2146 %:%589=633%:%

  2066 %:%568=627%:%

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  2067 %:%569=628%:%

  2148 %:%591=635%:%

  2068 %:%570=629%:%

  2149 %:%592=636%:%

  2069 %:%571=630%:%

  2150 %:%593=637%:%

  2070 %:%572=631%:%

  2151 %:%594=638%:%

  2071 %:%573=632%:%

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  2072 %:%574=633%:%

  2153 %:%596=640%:%

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  2154 %:%597=641%:%

  2074 %:%576=635%:%

  2155 %:%598=642%:%

  2075 %:%577=636%:%

  2156 %:%599=643%:%

  2076 %:%578=637%:%

  2157 %:%600=644%:%

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  2078 %:%580=639%:%

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  2080 %:%582=641%:%

  2161 %:%604=648%:%

  2081 %:%583=642%:%

  2162 %:%605=649%:%

  2082 %:%584=643%:%

  2163 %:%606=650%:%

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  2165 %:%608=652%:%

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  2166 %:%609=653%:%

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  2167 %:%610=654%:%

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  2168 %:%611=655%:%

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  2169 %:%612=656%:%

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  2170 %:%613=657%:%

  2090 %:%592=651%:%

  2171 %:%614=658%:%

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