author | wenzelm |
Mon, 30 Jul 2012 17:25:45 +0200 | |
changeset 48611 | b34ff75c23a7 |
parent 48519 | 5deda0549f97 |
permissions | -rw-r--r-- |
10602 | 1 |
% |
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\begin{isabellebody}% |
3 |
\def\isabellecontext{Numbers}% |
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% |
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\isadelimtheory |
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% |
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\endisadelimtheory |
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% |
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\isatagtheory |
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\isacommand{theory}\isamarkupfalse% |
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\ Numbers\isanewline |
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\isakeyword{imports}\ Complex{\isaliteral{5F}{\isacharunderscore}}Main\isanewline |
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\isakeyword{begin}% |
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\endisatagtheory |
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{\isafoldtheory}% |
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% |
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\isadelimtheory |
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% |
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\endisadelimtheory |
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% |
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\begin{isamarkuptext}% |
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numeric literals; default simprules; can re-orient% |
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\end{isamarkuptext}% |
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\isamarkuptrue% |
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\isacommand{lemma}\isamarkupfalse% |
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\ {\isaliteral{22}{\isachardoublequoteopen}}{\isadigit{2}}\ {\isaliteral{2A}{\isacharasterisk}}\ m\ {\isaliteral{3D}{\isacharequal}}\ m\ {\isaliteral{2B}{\isacharplus}}\ m{\isaliteral{22}{\isachardoublequoteclose}}% |
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\isadelimproof |
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% |
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\endisadelimproof |
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% |
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\isatagproof |
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% |
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\begin{isamarkuptxt}% |
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\begin{isabelle}% |
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\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{2}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2A}{\isacharasterisk}}\ m\ {\isaliteral{3D}{\isacharequal}}\ m\ {\isaliteral{2B}{\isacharplus}}\ m% |
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\end{isabelle}% |
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\end{isamarkuptxt}% |
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\isamarkuptrue% |
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\isacommand{oops}\isamarkupfalse% |
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% |
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\endisatagproof |
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{\isafoldproof}% |
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% |
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\isadelimproof |
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% |
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\endisadelimproof |
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\isanewline |
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\isanewline |
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\isacommand{fun}\isamarkupfalse% |
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\ h\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline |
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{\isaliteral{22}{\isachardoublequoteopen}}h\ i\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ i\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{3}}\ then\ {\isadigit{2}}\ else\ i{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}% |
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\begin{isamarkuptext}% |
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\isa{h\ {\isadigit{3}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{2}}} |
54 |
\isa{h\ i\ {\isaliteral{3D}{\isacharequal}}\ i}% |
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\end{isamarkuptext}% |
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\isamarkuptrue% |
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% |
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\begin{isamarkuptext}% |
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\begin{isabelle}% |
|
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Numeral{\isadigit{1}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{1}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}% |
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\end{isabelle} |
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\rulename{numeral_1_eq_1} |
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|
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\begin{isabelle}% |
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{\isadigit{2}}\ {\isaliteral{2B}{\isacharplus}}\ n\ {\isaliteral{3D}{\isacharequal}}\ Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}% |
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\end{isabelle} |
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\rulename{add_2_eq_Suc} |
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|
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\begin{isabelle}% |
|
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n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}% |
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\end{isabelle} |
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\rulename{add_2_eq_Suc'} |
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||
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\begin{isabelle}% |
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a\ {\isaliteral{2B}{\isacharplus}}\ b\ {\isaliteral{2B}{\isacharplus}}\ c\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}b\ {\isaliteral{2B}{\isacharplus}}\ c{\isaliteral{29}{\isacharparenright}}% |
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\end{isabelle} |
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\rulename{add_assoc} |
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|
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\begin{isabelle}% |
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a\ {\isaliteral{2B}{\isacharplus}}\ b\ {\isaliteral{3D}{\isacharequal}}\ b\ {\isaliteral{2B}{\isacharplus}}\ a% |
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\end{isabelle} |
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\rulename{add_commute} |
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|
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\begin{isabelle}% |
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b\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{2B}{\isacharplus}}\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}b\ {\isaliteral{2B}{\isacharplus}}\ c{\isaliteral{29}{\isacharparenright}}% |
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\end{isabelle} |
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\rulename{add_left_commute} |
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|
89 |
these form add_ac; similarly there is mult_ac% |
|
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\end{isamarkuptext}% |
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\isamarkuptrue% |
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\isacommand{lemma}\isamarkupfalse% |
|
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\ {\isaliteral{22}{\isachardoublequoteopen}}Suc{\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{2B}{\isacharplus}}\ j{\isaliteral{2A}{\isacharasterisk}}l{\isaliteral{2A}{\isacharasterisk}}k\ {\isaliteral{2B}{\isacharplus}}\ m{\isaliteral{2A}{\isacharasterisk}}n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ f\ {\isaliteral{28}{\isacharparenleft}}n{\isaliteral{2A}{\isacharasterisk}}m\ {\isaliteral{2B}{\isacharplus}}\ i\ {\isaliteral{2B}{\isacharplus}}\ k{\isaliteral{2A}{\isacharasterisk}}j{\isaliteral{2A}{\isacharasterisk}}l{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}% |
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\isadelimproof |
95 |
% |
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\endisadelimproof |
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% |
|
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\isatagproof |
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% |
100 |
\begin{isamarkuptxt}% |
|
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\begin{isabelle}% |
|
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\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ Suc\ {\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{2B}{\isacharplus}}\ j\ {\isaliteral{2A}{\isacharasterisk}}\ l\ {\isaliteral{2A}{\isacharasterisk}}\ k\ {\isaliteral{2B}{\isacharplus}}\ m\ {\isaliteral{2A}{\isacharasterisk}}\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ f\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2A}{\isacharasterisk}}\ m\ {\isaliteral{2B}{\isacharplus}}\ i\ {\isaliteral{2B}{\isacharplus}}\ k\ {\isaliteral{2A}{\isacharasterisk}}\ j\ {\isaliteral{2A}{\isacharasterisk}}\ l{\isaliteral{29}{\isacharparenright}}% |
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\end{isabelle}% |
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\end{isamarkuptxt}% |
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\isamarkuptrue% |
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\isacommand{apply}\isamarkupfalse% |
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\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ add{\isaliteral{5F}{\isacharunderscore}}ac\ mult{\isaliteral{5F}{\isacharunderscore}}ac{\isaliteral{29}{\isacharparenright}}% |
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\begin{isamarkuptxt}% |
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\begin{isabelle}% |
|
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\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ Suc\ {\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}m\ {\isaliteral{2A}{\isacharasterisk}}\ n\ {\isaliteral{2B}{\isacharplus}}\ j\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}k\ {\isaliteral{2A}{\isacharasterisk}}\ l{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\isanewline |
111 |
\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ }f\ {\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}m\ {\isaliteral{2A}{\isacharasterisk}}\ n\ {\isaliteral{2B}{\isacharplus}}\ j\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}k\ {\isaliteral{2A}{\isacharasterisk}}\ l{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}% |
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\end{isabelle}% |
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\end{isamarkuptxt}% |
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\isamarkuptrue% |
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\isacommand{oops}\isamarkupfalse% |
|
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% |
|
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\endisatagproof |
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{\isafoldproof}% |
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% |
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\isadelimproof |
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% |
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122 |
\endisadelimproof |
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% |
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\begin{isamarkuptext}% |
125 |
\begin{isabelle}% |
|
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m\ {\isaliteral{5C3C6C653E}{\isasymle}}\ n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ m\ div\ k\ {\isaliteral{5C3C6C653E}{\isasymle}}\ n\ div\ k% |
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\end{isabelle} |
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\rulename{div_le_mono} |
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|
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\begin{isabelle}% |
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{\isaliteral{28}{\isacharparenleft}}m\ {\isaliteral{2D}{\isacharminus}}\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2A}{\isacharasterisk}}\ k\ {\isaliteral{3D}{\isacharequal}}\ m\ {\isaliteral{2A}{\isacharasterisk}}\ k\ {\isaliteral{2D}{\isacharminus}}\ n\ {\isaliteral{2A}{\isacharasterisk}}\ k% |
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\end{isabelle} |
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\rulename{diff_mult_distrib} |
10602 | 134 |
|
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\begin{isabelle}% |
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a\ mod\ b\ {\isaliteral{2A}{\isacharasterisk}}\ c\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{2A}{\isacharasterisk}}\ c\ mod\ {\isaliteral{28}{\isacharparenleft}}b\ {\isaliteral{2A}{\isacharasterisk}}\ c{\isaliteral{29}{\isacharparenright}}% |
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\end{isabelle} |
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\rulename{mult_mod_left} |
10602 | 139 |
|
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\begin{isabelle}% |
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P\ {\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{2D}{\isacharminus}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{3C}{\isacharless}}\ b\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ P\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}d{\isaliteral{2E}{\isachardot}}\ a\ {\isaliteral{3D}{\isacharequal}}\ b\ {\isaliteral{2B}{\isacharplus}}\ d\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ P\ d{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}% |
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\end{isabelle} |
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\rulename{nat_diff_split}% |
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\end{isamarkuptext}% |
|
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\isamarkuptrue% |
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\isacommand{lemma}\isamarkupfalse% |
|
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\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ n\ {\isaliteral{2A}{\isacharasterisk}}\ n\ {\isaliteral{2D}{\isacharminus}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{1}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}nat{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline |
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% |
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\isadelimproof |
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% |
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\endisadelimproof |
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% |
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\isatagproof |
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\isacommand{apply}\isamarkupfalse% |
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\ {\isaliteral{28}{\isacharparenleft}}clarsimp\ split{\isaliteral{3A}{\isacharcolon}}\ nat{\isaliteral{5F}{\isacharunderscore}}diff{\isaliteral{5F}{\isacharunderscore}}split\ iff\ del{\isaliteral{3A}{\isacharcolon}}\ less{\isaliteral{5F}{\isacharunderscore}}Suc{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline |
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\ % |
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\isamarkupcmt{\begin{isabelle}% |
|
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\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}d{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}n\ {\isaliteral{3C}{\isacharless}}\ Suc\ {\isadigit{0}}{\isaliteral{3B}{\isacharsemicolon}}\ n\ {\isaliteral{2A}{\isacharasterisk}}\ n\ {\isaliteral{3D}{\isacharequal}}\ Suc\ d{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ d\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}% |
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\end{isabelle}% |
160 |
} |
|
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\isanewline |
|
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\isacommand{apply}\isamarkupfalse% |
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\ {\isaliteral{28}{\isacharparenleft}}subgoal{\isaliteral{5F}{\isacharunderscore}}tac\ {\isaliteral{22}{\isachardoublequoteopen}}n{\isaliteral{3D}{\isacharequal}}{\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{2C}{\isacharcomma}}\ force{\isaliteral{2C}{\isacharcomma}}\ arith{\isaliteral{29}{\isacharparenright}}\isanewline |
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\isacommand{done}\isamarkupfalse% |
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% |
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\endisatagproof |
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{\isafoldproof}% |
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% |
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\isadelimproof |
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\isanewline |
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% |
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\endisadelimproof |
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12156
d2758965362e
new-style numerals without leading #, along with generic 0 and 1
paulson
parents:
11870
diff
changeset
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\isanewline |
d2758965362e
new-style numerals without leading #, along with generic 0 and 1
paulson
parents:
11870
diff
changeset
|
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\isanewline |
17175 | 175 |
\isacommand{lemma}\isamarkupfalse% |
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\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ n\ {\isaliteral{2A}{\isacharasterisk}}\ n\ {\isaliteral{2D}{\isacharminus}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{4}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}nat{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline |
17056 | 177 |
% |
178 |
\isadelimproof |
|
179 |
% |
|
180 |
\endisadelimproof |
|
181 |
% |
|
182 |
\isatagproof |
|
17175 | 183 |
\isacommand{apply}\isamarkupfalse% |
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\ {\isaliteral{28}{\isacharparenleft}}simp\ split{\isaliteral{3A}{\isacharcolon}}\ nat{\isaliteral{5F}{\isacharunderscore}}diff{\isaliteral{5F}{\isacharunderscore}}split{\isaliteral{2C}{\isacharcomma}}\ clarify{\isaliteral{29}{\isacharparenright}}\isanewline |
16353 | 185 |
\ % |
186 |
\isamarkupcmt{\begin{isabelle}% |
|
40406 | 187 |
\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}d{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}n\ {\isaliteral{3C}{\isacharless}}\ {\isadigit{2}}{\isaliteral{3B}{\isacharsemicolon}}\ n\ {\isaliteral{2A}{\isacharasterisk}}\ n\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{4}}\ {\isaliteral{2B}{\isacharplus}}\ d{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ d\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}% |
16353 | 188 |
\end{isabelle}% |
189 |
} |
|
190 |
\isanewline |
|
17175 | 191 |
\isacommand{apply}\isamarkupfalse% |
40406 | 192 |
\ {\isaliteral{28}{\isacharparenleft}}subgoal{\isaliteral{5F}{\isacharunderscore}}tac\ {\isaliteral{22}{\isachardoublequoteopen}}n{\isaliteral{3D}{\isacharequal}}{\isadigit{0}}\ {\isaliteral{7C}{\isacharbar}}\ n{\isaliteral{3D}{\isacharequal}}{\isadigit{1}}{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{2C}{\isacharcomma}}\ force{\isaliteral{2C}{\isacharcomma}}\ arith{\isaliteral{29}{\isacharparenright}}\isanewline |
17175 | 193 |
\isacommand{done}\isamarkupfalse% |
194 |
% |
|
17056 | 195 |
\endisatagproof |
196 |
{\isafoldproof}% |
|
197 |
% |
|
198 |
\isadelimproof |
|
199 |
% |
|
200 |
\endisadelimproof |
|
11866 | 201 |
% |
11187 | 202 |
\begin{isamarkuptext}% |
203 |
\begin{isabelle}% |
|
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m\ mod\ n\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ m\ {\isaliteral{3C}{\isacharless}}\ n\ then\ m\ else\ {\isaliteral{28}{\isacharparenleft}}m\ {\isaliteral{2D}{\isacharminus}}\ n{\isaliteral{29}{\isacharparenright}}\ mod\ n{\isaliteral{29}{\isacharparenright}}% |
11187 | 205 |
\end{isabelle} |
11174 | 206 |
\rulename{mod_if} |
11187 | 207 |
|
208 |
\begin{isabelle}% |
|
40406 | 209 |
a\ div\ b\ {\isaliteral{2A}{\isacharasterisk}}\ b\ {\isaliteral{2B}{\isacharplus}}\ a\ mod\ b\ {\isaliteral{3D}{\isacharequal}}\ a% |
10602 | 210 |
\end{isabelle} |
11174 | 211 |
\rulename{mod_div_equality} |
10602 | 212 |
|
213 |
||
11187 | 214 |
\begin{isabelle}% |
40406 | 215 |
a\ {\isaliteral{2A}{\isacharasterisk}}\ b\ div\ c\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}b\ div\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2B}{\isacharplus}}\ a\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}b\ mod\ c{\isaliteral{29}{\isacharparenright}}\ div\ c% |
11187 | 216 |
\end{isabelle} |
11174 | 217 |
\rulename{div_mult1_eq} |
10602 | 218 |
|
11187 | 219 |
\begin{isabelle}% |
40406 | 220 |
a\ {\isaliteral{2A}{\isacharasterisk}}\ b\ mod\ c\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}b\ mod\ c{\isaliteral{29}{\isacharparenright}}\ mod\ c% |
11187 | 221 |
\end{isabelle} |
30224 | 222 |
\rulename{mod_mult_right_eq} |
10602 | 223 |
|
11187 | 224 |
\begin{isabelle}% |
40406 | 225 |
a\ div\ {\isaliteral{28}{\isacharparenleft}}b\ {\isaliteral{2A}{\isacharasterisk}}\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ a\ div\ b\ div\ c% |
11187 | 226 |
\end{isabelle} |
11174 | 227 |
\rulename{div_mult2_eq} |
10602 | 228 |
|
11187 | 229 |
\begin{isabelle}% |
40406 | 230 |
a\ mod\ {\isaliteral{28}{\isacharparenleft}}b\ {\isaliteral{2A}{\isacharasterisk}}\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ b\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}a\ div\ b\ mod\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2B}{\isacharplus}}\ a\ mod\ b% |
11187 | 231 |
\end{isabelle} |
11174 | 232 |
\rulename{mod_mult2_eq} |
10602 | 233 |
|
11187 | 234 |
\begin{isabelle}% |
40406 | 235 |
c\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ c\ {\isaliteral{2A}{\isacharasterisk}}\ a\ div\ {\isaliteral{28}{\isacharparenleft}}c\ {\isaliteral{2A}{\isacharasterisk}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ a\ div\ b% |
11187 | 236 |
\end{isabelle} |
11174 | 237 |
\rulename{div_mult_mult1} |
10602 | 238 |
|
11187 | 239 |
\begin{isabelle}% |
40406 | 240 |
a\ div\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}% |
11187 | 241 |
\end{isabelle} |
27658 | 242 |
\rulename{div_by_0} |
10602 | 243 |
|
11187 | 244 |
\begin{isabelle}% |
40406 | 245 |
a\ mod\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ a% |
11187 | 246 |
\end{isabelle} |
27658 | 247 |
\rulename{mod_by_0} |
10602 | 248 |
|
11187 | 249 |
\begin{isabelle}% |
40406 | 250 |
{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}m\ dvd\ n{\isaliteral{3B}{\isacharsemicolon}}\ n\ dvd\ m{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ m\ {\isaliteral{3D}{\isacharequal}}\ n% |
11187 | 251 |
\end{isabelle} |
33750 | 252 |
\rulename{dvd_antisym} |
10602 | 253 |
|
11187 | 254 |
\begin{isabelle}% |
40406 | 255 |
{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}a\ dvd\ b{\isaliteral{3B}{\isacharsemicolon}}\ a\ dvd\ c{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ a\ dvd\ b\ {\isaliteral{2B}{\isacharplus}}\ c% |
11187 | 256 |
\end{isabelle} |
11174 | 257 |
\rulename{dvd_add} |
10602 | 258 |
|
259 |
For the integers, I'd list a few theorems that somehow involve negative |
|
13758 | 260 |
numbers.% |
261 |
\end{isamarkuptext}% |
|
262 |
\isamarkuptrue% |
|
263 |
% |
|
264 |
\begin{isamarkuptext}% |
|
10602 | 265 |
Division, remainder of negatives |
266 |
||
267 |
||
11187 | 268 |
\begin{isabelle}% |
40406 | 269 |
{\isadigit{0}}\ {\isaliteral{3C}{\isacharless}}\ b\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isadigit{0}}\ {\isaliteral{5C3C6C653E}{\isasymle}}\ a\ mod\ b% |
11187 | 270 |
\end{isabelle} |
11174 | 271 |
\rulename{pos_mod_sign} |
10602 | 272 |
|
11187 | 273 |
\begin{isabelle}% |
40406 | 274 |
{\isadigit{0}}\ {\isaliteral{3C}{\isacharless}}\ b\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ a\ mod\ b\ {\isaliteral{3C}{\isacharless}}\ b% |
11174 | 275 |
\end{isabelle} |
11187 | 276 |
\rulename{pos_mod_bound} |
10602 | 277 |
|
11187 | 278 |
\begin{isabelle}% |
40406 | 279 |
b\ {\isaliteral{3C}{\isacharless}}\ {\isadigit{0}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ a\ mod\ b\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isadigit{0}}% |
11174 | 280 |
\end{isabelle} |
11187 | 281 |
\rulename{neg_mod_sign} |
10602 | 282 |
|
11187 | 283 |
\begin{isabelle}% |
40406 | 284 |
b\ {\isaliteral{3C}{\isacharless}}\ {\isadigit{0}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ b\ {\isaliteral{3C}{\isacharless}}\ a\ mod\ b% |
11187 | 285 |
\end{isabelle} |
11174 | 286 |
\rulename{neg_mod_bound} |
10602 | 287 |
|
11187 | 288 |
\begin{isabelle}% |
40406 | 289 |
{\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{2B}{\isacharplus}}\ b{\isaliteral{29}{\isacharparenright}}\ div\ c\ {\isaliteral{3D}{\isacharequal}}\ a\ div\ c\ {\isaliteral{2B}{\isacharplus}}\ b\ div\ c\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}a\ mod\ c\ {\isaliteral{2B}{\isacharplus}}\ b\ mod\ c{\isaliteral{29}{\isacharparenright}}\ div\ c% |
11174 | 290 |
\end{isabelle} |
11187 | 291 |
\rulename{zdiv_zadd1_eq} |
10602 | 292 |
|
11187 | 293 |
\begin{isabelle}% |
40406 | 294 |
{\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{2B}{\isacharplus}}\ b{\isaliteral{29}{\isacharparenright}}\ mod\ c\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}a\ mod\ c\ {\isaliteral{2B}{\isacharplus}}\ b\ mod\ c{\isaliteral{29}{\isacharparenright}}\ mod\ c% |
11187 | 295 |
\end{isabelle} |
30224 | 296 |
\rulename{mod_add_eq} |
10602 | 297 |
|
11187 | 298 |
\begin{isabelle}% |
40406 | 299 |
a\ {\isaliteral{2A}{\isacharasterisk}}\ b\ div\ c\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}b\ div\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2B}{\isacharplus}}\ a\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}b\ mod\ c{\isaliteral{29}{\isacharparenright}}\ div\ c% |
10602 | 300 |
\end{isabelle} |
11187 | 301 |
\rulename{zdiv_zmult1_eq} |
10602 | 302 |
|
11187 | 303 |
\begin{isabelle}% |
40406 | 304 |
a\ {\isaliteral{2A}{\isacharasterisk}}\ b\ mod\ c\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}b\ mod\ c{\isaliteral{29}{\isacharparenright}}\ mod\ c% |
11187 | 305 |
\end{isabelle} |
47183 | 306 |
\rulename{mod_mult_right_eq} |
10602 | 307 |
|
11187 | 308 |
\begin{isabelle}% |
40406 | 309 |
{\isadigit{0}}\ {\isaliteral{3C}{\isacharless}}\ c\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ a\ div\ {\isaliteral{28}{\isacharparenleft}}b\ {\isaliteral{2A}{\isacharasterisk}}\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ a\ div\ b\ div\ c% |
11174 | 310 |
\end{isabelle} |
11187 | 311 |
\rulename{zdiv_zmult2_eq} |
10602 | 312 |
|
11187 | 313 |
\begin{isabelle}% |
40406 | 314 |
{\isadigit{0}}\ {\isaliteral{3C}{\isacharless}}\ c\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ a\ mod\ {\isaliteral{28}{\isacharparenleft}}b\ {\isaliteral{2A}{\isacharasterisk}}\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ b\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}a\ div\ b\ mod\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2B}{\isacharplus}}\ a\ mod\ b% |
11187 | 315 |
\end{isabelle} |
14400 | 316 |
\rulename{zmod_zmult2_eq}% |
11187 | 317 |
\end{isamarkuptext}% |
17175 | 318 |
\isamarkuptrue% |
319 |
\isacommand{lemma}\isamarkupfalse% |
|
40406 | 320 |
\ {\isaliteral{22}{\isachardoublequoteopen}}abs\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2B}{\isacharplus}}y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6C653E}{\isasymle}}\ abs\ x\ {\isaliteral{2B}{\isacharplus}}\ abs\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline |
17056 | 321 |
% |
322 |
\isadelimproof |
|
323 |
% |
|
324 |
\endisadelimproof |
|
325 |
% |
|
326 |
\isatagproof |
|
17175 | 327 |
\isacommand{by}\isamarkupfalse% |
328 |
\ arith% |
|
17056 | 329 |
\endisatagproof |
330 |
{\isafoldproof}% |
|
331 |
% |
|
332 |
\isadelimproof |
|
333 |
\isanewline |
|
334 |
% |
|
335 |
\endisadelimproof |
|
15481 | 336 |
\isanewline |
17175 | 337 |
\isacommand{lemma}\isamarkupfalse% |
40406 | 338 |
\ {\isaliteral{22}{\isachardoublequoteopen}}abs\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{2}}{\isaliteral{2A}{\isacharasterisk}}x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{2}}\ {\isaliteral{2A}{\isacharasterisk}}\ abs\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline |
17056 | 339 |
% |
340 |
\isadelimproof |
|
341 |
% |
|
342 |
\endisadelimproof |
|
343 |
% |
|
344 |
\isatagproof |
|
17175 | 345 |
\isacommand{by}\isamarkupfalse% |
40406 | 346 |
\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ abs{\isaliteral{5F}{\isacharunderscore}}if{\isaliteral{29}{\isacharparenright}}% |
17056 | 347 |
\endisatagproof |
348 |
{\isafoldproof}% |
|
349 |
% |
|
350 |
\isadelimproof |
|
351 |
% |
|
352 |
\endisadelimproof |
|
11866 | 353 |
% |
11187 | 354 |
\begin{isamarkuptext}% |
13758 | 355 |
Induction rules for the Integers |
356 |
||
357 |
\begin{isabelle}% |
|
40406 | 358 |
{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}k\ {\isaliteral{5C3C6C653E}{\isasymle}}\ i{\isaliteral{3B}{\isacharsemicolon}}\ P\ k{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}i{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}k\ {\isaliteral{5C3C6C653E}{\isasymle}}\ i{\isaliteral{3B}{\isacharsemicolon}}\ P\ i{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ {\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ i% |
13758 | 359 |
\end{isabelle} |
360 |
\rulename{int_ge_induct} |
|
361 |
||
362 |
\begin{isabelle}% |
|
40406 | 363 |
{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}k\ {\isaliteral{3C}{\isacharless}}\ i{\isaliteral{3B}{\isacharsemicolon}}\ P\ {\isaliteral{28}{\isacharparenleft}}k\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}i{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}k\ {\isaliteral{3C}{\isacharless}}\ i{\isaliteral{3B}{\isacharsemicolon}}\ P\ i{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ {\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ i% |
13758 | 364 |
\end{isabelle} |
365 |
\rulename{int_gr_induct} |
|
366 |
||
367 |
\begin{isabelle}% |
|
40406 | 368 |
{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}i\ {\isaliteral{5C3C6C653E}{\isasymle}}\ k{\isaliteral{3B}{\isacharsemicolon}}\ P\ k{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}i{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}i\ {\isaliteral{5C3C6C653E}{\isasymle}}\ k{\isaliteral{3B}{\isacharsemicolon}}\ P\ i{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ {\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ i% |
13758 | 369 |
\end{isabelle} |
370 |
\rulename{int_le_induct} |
|
371 |
||
372 |
\begin{isabelle}% |
|
40406 | 373 |
{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}i\ {\isaliteral{3C}{\isacharless}}\ k{\isaliteral{3B}{\isacharsemicolon}}\ P\ {\isaliteral{28}{\isacharparenleft}}k\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}i{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}i\ {\isaliteral{3C}{\isacharless}}\ k{\isaliteral{3B}{\isacharsemicolon}}\ P\ i{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ {\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ i% |
13758 | 374 |
\end{isabelle} |
375 |
\rulename{int_less_induct}% |
|
376 |
\end{isamarkuptext}% |
|
377 |
\isamarkuptrue% |
|
378 |
% |
|
379 |
\begin{isamarkuptext}% |
|
14400 | 380 |
FIELDS |
10776 | 381 |
|
11187 | 382 |
\begin{isabelle}% |
40406 | 383 |
x\ {\isaliteral{3C}{\isacharless}}\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}z{\isaliteral{3E}{\isachargreater}}x{\isaliteral{2E}{\isachardot}}\ z\ {\isaliteral{3C}{\isacharless}}\ y% |
11187 | 384 |
\end{isabelle} |
14295 | 385 |
\rulename{dense} |
10776 | 386 |
|
11187 | 387 |
\begin{isabelle}% |
40406 | 388 |
a\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}b\ {\isaliteral{2F}{\isacharslash}}\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{2A}{\isacharasterisk}}\ b\ {\isaliteral{2F}{\isacharslash}}\ c% |
11187 | 389 |
\end{isabelle} |
14288 | 390 |
\rulename{times_divide_eq_right} |
10776 | 391 |
|
11187 | 392 |
\begin{isabelle}% |
40406 | 393 |
b\ {\isaliteral{2F}{\isacharslash}}\ c\ {\isaliteral{2A}{\isacharasterisk}}\ a\ {\isaliteral{3D}{\isacharequal}}\ b\ {\isaliteral{2A}{\isacharasterisk}}\ a\ {\isaliteral{2F}{\isacharslash}}\ c% |
11187 | 394 |
\end{isabelle} |
14288 | 395 |
\rulename{times_divide_eq_left} |
10776 | 396 |
|
11187 | 397 |
\begin{isabelle}% |
40406 | 398 |
a\ {\isaliteral{2F}{\isacharslash}}\ {\isaliteral{28}{\isacharparenleft}}b\ {\isaliteral{2F}{\isacharslash}}\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{2A}{\isacharasterisk}}\ c\ {\isaliteral{2F}{\isacharslash}}\ b% |
11187 | 399 |
\end{isabelle} |
14288 | 400 |
\rulename{divide_divide_eq_right} |
10776 | 401 |
|
11187 | 402 |
\begin{isabelle}% |
40406 | 403 |
a\ {\isaliteral{2F}{\isacharslash}}\ b\ {\isaliteral{2F}{\isacharslash}}\ c\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{2F}{\isacharslash}}\ {\isaliteral{28}{\isacharparenleft}}b\ {\isaliteral{2A}{\isacharasterisk}}\ c{\isaliteral{29}{\isacharparenright}}% |
11187 | 404 |
\end{isabelle} |
14288 | 405 |
\rulename{divide_divide_eq_left} |
10776 | 406 |
|
11187 | 407 |
\begin{isabelle}% |
40406 | 408 |
{\isaliteral{2D}{\isacharminus}}\ {\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{2F}{\isacharslash}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{2D}{\isacharminus}}\ a\ {\isaliteral{2F}{\isacharslash}}\ b% |
11187 | 409 |
\end{isabelle} |
14295 | 410 |
\rulename{minus_divide_left} |
10776 | 411 |
|
11187 | 412 |
\begin{isabelle}% |
40406 | 413 |
{\isaliteral{2D}{\isacharminus}}\ {\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{2F}{\isacharslash}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{2F}{\isacharslash}}\ {\isaliteral{2D}{\isacharminus}}\ b% |
11187 | 414 |
\end{isabelle} |
14295 | 415 |
\rulename{minus_divide_right} |
10776 | 416 |
|
417 |
This last NOT a simprule |
|
418 |
||
11187 | 419 |
\begin{isabelle}% |
40406 | 420 |
{\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{2B}{\isacharplus}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2F}{\isacharslash}}\ c\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{2F}{\isacharslash}}\ c\ {\isaliteral{2B}{\isacharplus}}\ b\ {\isaliteral{2F}{\isacharslash}}\ c% |
11174 | 421 |
\end{isabelle} |
14295 | 422 |
\rulename{add_divide_distrib}% |
11187 | 423 |
\end{isamarkuptext}% |
17175 | 424 |
\isamarkuptrue% |
425 |
\isacommand{lemma}\isamarkupfalse% |
|
40406 | 426 |
\ {\isaliteral{22}{\isachardoublequoteopen}}{\isadigit{3}}{\isaliteral{2F}{\isacharslash}}{\isadigit{4}}\ {\isaliteral{3C}{\isacharless}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{7}}{\isaliteral{2F}{\isacharslash}}{\isadigit{8}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ real{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline |
17056 | 427 |
% |
428 |
\isadelimproof |
|
429 |
% |
|
430 |
\endisadelimproof |
|
431 |
% |
|
432 |
\isatagproof |
|
17175 | 433 |
\isacommand{by}\isamarkupfalse% |
434 |
\ simp% |
|
17056 | 435 |
\endisatagproof |
436 |
{\isafoldproof}% |
|
437 |
% |
|
438 |
\isadelimproof |
|
439 |
\ \isanewline |
|
440 |
% |
|
441 |
\endisadelimproof |
|
11174 | 442 |
\isanewline |
17175 | 443 |
\isacommand{lemma}\isamarkupfalse% |
40406 | 444 |
\ {\isaliteral{22}{\isachardoublequoteopen}}P\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}{\isadigit{3}}{\isaliteral{2F}{\isacharslash}}{\isadigit{4}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{8}}{\isaliteral{2F}{\isacharslash}}{\isadigit{1}}{\isadigit{5}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ real{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}% |
17056 | 445 |
\isadelimproof |
446 |
% |
|
447 |
\endisadelimproof |
|
448 |
% |
|
449 |
\isatagproof |
|
16353 | 450 |
% |
451 |
\begin{isamarkuptxt}% |
|
452 |
\begin{isabelle}% |
|
40406 | 453 |
\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ P\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{3}}\ {\isaliteral{2F}{\isacharslash}}\ {\isadigit{4}}\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{8}}\ {\isaliteral{2F}{\isacharslash}}\ {\isadigit{1}}{\isadigit{5}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}% |
16353 | 454 |
\end{isabelle}% |
455 |
\end{isamarkuptxt}% |
|
17175 | 456 |
\isamarkuptrue% |
457 |
\isacommand{apply}\isamarkupfalse% |
|
458 |
\ simp% |
|
16353 | 459 |
\begin{isamarkuptxt}% |
460 |
\begin{isabelle}% |
|
40406 | 461 |
\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ P\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{2}}\ {\isaliteral{2F}{\isacharslash}}\ {\isadigit{5}}{\isaliteral{29}{\isacharparenright}}% |
16353 | 462 |
\end{isabelle}% |
463 |
\end{isamarkuptxt}% |
|
17175 | 464 |
\isamarkuptrue% |
465 |
\isacommand{oops}\isamarkupfalse% |
|
466 |
% |
|
17056 | 467 |
\endisatagproof |
468 |
{\isafoldproof}% |
|
469 |
% |
|
470 |
\isadelimproof |
|
471 |
% |
|
472 |
\endisadelimproof |
|
473 |
\isanewline |
|
15481 | 474 |
\isanewline |
17175 | 475 |
\isacommand{lemma}\isamarkupfalse% |
40406 | 476 |
\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isadigit{3}}{\isaliteral{2F}{\isacharslash}}{\isadigit{4}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{8}}{\isaliteral{2F}{\isacharslash}}{\isadigit{1}}{\isadigit{5}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3C}{\isacharless}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ real{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}% |
17056 | 477 |
\isadelimproof |
478 |
% |
|
479 |
\endisadelimproof |
|
480 |
% |
|
481 |
\isatagproof |
|
16353 | 482 |
% |
483 |
\begin{isamarkuptxt}% |
|
484 |
\begin{isabelle}% |
|
40406 | 485 |
\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isadigit{3}}\ {\isaliteral{2F}{\isacharslash}}\ {\isadigit{4}}\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{8}}\ {\isaliteral{2F}{\isacharslash}}\ {\isadigit{1}}{\isadigit{5}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3C}{\isacharless}}\ x% |
16353 | 486 |
\end{isabelle}% |
487 |
\end{isamarkuptxt}% |
|
17175 | 488 |
\isamarkuptrue% |
489 |
\isacommand{apply}\isamarkupfalse% |
|
490 |
\ simp% |
|
16353 | 491 |
\begin{isamarkuptxt}% |
492 |
\begin{isabelle}% |
|
40406 | 493 |
\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isadigit{2}}\ {\isaliteral{3C}{\isacharless}}\ x\ {\isaliteral{2A}{\isacharasterisk}}\ {\isadigit{5}}% |
16353 | 494 |
\end{isabelle}% |
495 |
\end{isamarkuptxt}% |
|
17175 | 496 |
\isamarkuptrue% |
497 |
\isacommand{oops}\isamarkupfalse% |
|
498 |
% |
|
17056 | 499 |
\endisatagproof |
500 |
{\isafoldproof}% |
|
501 |
% |
|
502 |
\isadelimproof |
|
503 |
% |
|
504 |
\endisadelimproof |
|
14400 | 505 |
% |
506 |
\begin{isamarkuptext}% |
|
507 |
Ring and Field |
|
508 |
||
509 |
Requires a field, or else an ordered ring |
|
510 |
||
511 |
\begin{isabelle}% |
|
40406 | 512 |
{\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{2A}{\isacharasterisk}}\ b\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ b\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}% |
14400 | 513 |
\end{isabelle} |
514 |
\rulename{mult_eq_0_iff} |
|
515 |
||
516 |
\begin{isabelle}% |
|
40406 | 517 |
{\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{2A}{\isacharasterisk}}\ c\ {\isaliteral{3D}{\isacharequal}}\ b\ {\isaliteral{2A}{\isacharasterisk}}\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}c\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ a\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{29}{\isacharparenright}}% |
14400 | 518 |
\end{isabelle} |
519 |
\rulename{mult_cancel_right} |
|
520 |
||
521 |
\begin{isabelle}% |
|
40406 | 522 |
{\isaliteral{28}{\isacharparenleft}}c\ {\isaliteral{2A}{\isacharasterisk}}\ a\ {\isaliteral{3D}{\isacharequal}}\ c\ {\isaliteral{2A}{\isacharasterisk}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}c\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ a\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{29}{\isacharparenright}}% |
14400 | 523 |
\end{isabelle} |
23504 | 524 |
\rulename{mult_cancel_left}% |
14400 | 525 |
\end{isamarkuptext}% |
17175 | 526 |
\isamarkuptrue% |
17056 | 527 |
% |
14400 | 528 |
\begin{isamarkuptext}% |
529 |
effect of show sorts on the above |
|
530 |
||
531 |
\begin{isabelle}% |
|
40406 | 532 |
{\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}c{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}ring{\isaliteral{5F}{\isacharunderscore}}no{\isaliteral{5F}{\isacharunderscore}}zero{\isaliteral{5F}{\isacharunderscore}}divisors{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}a{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}ring{\isaliteral{5F}{\isacharunderscore}}no{\isaliteral{5F}{\isacharunderscore}}zero{\isaliteral{5F}{\isacharunderscore}}divisors{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\isanewline |
533 |
\isaindent{{\isaliteral{28}{\isacharparenleft}}}c\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}b{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}ring{\isaliteral{5F}{\isacharunderscore}}no{\isaliteral{5F}{\isacharunderscore}}zero{\isaliteral{5F}{\isacharunderscore}}divisors{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\isanewline |
|
534 |
{\isaliteral{28}{\isacharparenleft}}c\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}ring{\isaliteral{5F}{\isacharunderscore}}no{\isaliteral{5F}{\isacharunderscore}}zero{\isaliteral{5F}{\isacharunderscore}}divisors{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ a\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{29}{\isacharparenright}}% |
|
14400 | 535 |
\end{isabelle} |
23504 | 536 |
\rulename{mult_cancel_left}% |
14400 | 537 |
\end{isamarkuptext}% |
17175 | 538 |
\isamarkuptrue% |
17056 | 539 |
% |
14400 | 540 |
\begin{isamarkuptext}% |
541 |
absolute value |
|
542 |
||
543 |
\begin{isabelle}% |
|
40406 | 544 |
{\isaliteral{5C3C6261723E}{\isasymbar}}a\ {\isaliteral{2A}{\isacharasterisk}}\ b{\isaliteral{5C3C6261723E}{\isasymbar}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C6261723E}{\isasymbar}}a{\isaliteral{5C3C6261723E}{\isasymbar}}\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{5C3C6261723E}{\isasymbar}}b{\isaliteral{5C3C6261723E}{\isasymbar}}% |
14400 | 545 |
\end{isabelle} |
546 |
\rulename{abs_mult} |
|
547 |
||
548 |
\begin{isabelle}% |
|
40406 | 549 |
{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6261723E}{\isasymbar}}a{\isaliteral{5C3C6261723E}{\isasymbar}}\ {\isaliteral{5C3C6C653E}{\isasymle}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{5C3C6C653E}{\isasymle}}\ b\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{2D}{\isacharminus}}\ a\ {\isaliteral{5C3C6C653E}{\isasymle}}\ b{\isaliteral{29}{\isacharparenright}}% |
14400 | 550 |
\end{isabelle} |
551 |
\rulename{abs_le_iff} |
|
552 |
||
553 |
\begin{isabelle}% |
|
40406 | 554 |
{\isaliteral{5C3C6261723E}{\isasymbar}}a\ {\isaliteral{2B}{\isacharplus}}\ b{\isaliteral{5C3C6261723E}{\isasymbar}}\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isaliteral{5C3C6261723E}{\isasymbar}}a{\isaliteral{5C3C6261723E}{\isasymbar}}\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{5C3C6261723E}{\isasymbar}}b{\isaliteral{5C3C6261723E}{\isasymbar}}% |
14400 | 555 |
\end{isabelle} |
556 |
\rulename{abs_triangle_ineq} |
|
557 |
||
558 |
\begin{isabelle}% |
|
40406 | 559 |
a\isaliteral{5C3C5E627375703E}{}\isactrlbsup m\ {\isaliteral{2B}{\isacharplus}}\ n\isaliteral{5C3C5E657375703E}{}\isactrlesup \ {\isaliteral{3D}{\isacharequal}}\ a\isaliteral{5C3C5E627375703E}{}\isactrlbsup m\isaliteral{5C3C5E657375703E}{}\isactrlesup \ {\isaliteral{2A}{\isacharasterisk}}\ a\isaliteral{5C3C5E627375703E}{}\isactrlbsup n\isaliteral{5C3C5E657375703E}{}\isactrlesup % |
14400 | 560 |
\end{isabelle} |
561 |
\rulename{power_add} |
|
562 |
||
563 |
\begin{isabelle}% |
|
40406 | 564 |
a\isaliteral{5C3C5E627375703E}{}\isactrlbsup m\ {\isaliteral{2A}{\isacharasterisk}}\ n\isaliteral{5C3C5E657375703E}{}\isactrlesup \ {\isaliteral{3D}{\isacharequal}}\ a\isaliteral{5C3C5E627375703E}{}\isactrlbsup m\isaliteral{5C3C5E657375703E}{}\isactrlesup \isaliteral{5C3C5E627375703E}{}\isactrlbsup n\isaliteral{5C3C5E657375703E}{}\isactrlesup % |
14400 | 565 |
\end{isabelle} |
566 |
\rulename{power_mult} |
|
567 |
||
568 |
\begin{isabelle}% |
|
40406 | 569 |
{\isaliteral{5C3C6261723E}{\isasymbar}}a\isaliteral{5C3C5E627375703E}{}\isactrlbsup n\isaliteral{5C3C5E657375703E}{}\isactrlesup {\isaliteral{5C3C6261723E}{\isasymbar}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C6261723E}{\isasymbar}}a{\isaliteral{5C3C6261723E}{\isasymbar}}\isaliteral{5C3C5E627375703E}{}\isactrlbsup n\isaliteral{5C3C5E657375703E}{}\isactrlesup % |
14400 | 570 |
\end{isabelle} |
571 |
\rulename{power_abs}% |
|
572 |
\end{isamarkuptext}% |
|
17175 | 573 |
\isamarkuptrue% |
17056 | 574 |
% |
575 |
\isadelimtheory |
|
576 |
% |
|
577 |
\endisadelimtheory |
|
578 |
% |
|
579 |
\isatagtheory |
|
17175 | 580 |
\isacommand{end}\isamarkupfalse% |
581 |
% |
|
17056 | 582 |
\endisatagtheory |
583 |
{\isafoldtheory}% |
|
584 |
% |
|
585 |
\isadelimtheory |
|
586 |
% |
|
587 |
\endisadelimtheory |
|
588 |
\isanewline |
|
11187 | 589 |
\end{isabellebody}% |
10602 | 590 |
%%% Local Variables: |
591 |
%%% mode: latex |
|
592 |
%%% TeX-master: "root" |
|
593 |
%%% End: |