diff -r 42671298f037 -r 313a24b66a8d doc-src/IsarRef/Thy/document/First_Order_Logic.tex --- a/doc-src/IsarRef/Thy/document/First_Order_Logic.tex Sun Nov 07 23:32:26 2010 +0100 +++ b/doc-src/IsarRef/Thy/document/First_Order_Logic.tex Mon Nov 08 00:00:47 2010 +0100 @@ -1,6 +1,6 @@ % \begin{isabellebody}% -\def\isabellecontext{First{\isacharunderscore}Order{\isacharunderscore}Logic}% +\def\isabellecontext{First{\isaliteral{5F}{\isacharunderscore}}Order{\isaliteral{5F}{\isacharunderscore}}Logic}% % \isamarkupheader{Example: First-Order Logic% } @@ -12,7 +12,7 @@ % \isatagvisible \isacommand{theory}\isamarkupfalse% -\ First{\isacharunderscore}Order{\isacharunderscore}Logic\isanewline +\ First{\isaliteral{5F}{\isacharunderscore}}Order{\isaliteral{5F}{\isacharunderscore}}Logic\isanewline \isakeyword{imports}\ Pure\isanewline \isakeyword{begin}% \endisatagvisible @@ -24,8 +24,8 @@ % \begin{isamarkuptext}% \noindent In order to commence a new object-logic within - Isabelle/Pure we introduce abstract syntactic categories \isa{{\isachardoublequote}i{\isachardoublequote}} - for individuals and \isa{{\isachardoublequote}o{\isachardoublequote}} for object-propositions. The latter + Isabelle/Pure we introduce abstract syntactic categories \isa{{\isaliteral{22}{\isachardoublequote}}i{\isaliteral{22}{\isachardoublequote}}} + for individuals and \isa{{\isaliteral{22}{\isachardoublequote}}o{\isaliteral{22}{\isachardoublequote}}} for object-propositions. The latter is embedded into the language of Pure propositions by means of a separate judgment.% \end{isamarkuptext}% @@ -37,10 +37,10 @@ \isanewline \isacommand{judgment}\isamarkupfalse% \isanewline -\ \ Trueprop\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}o\ {\isasymRightarrow}\ prop{\isachardoublequoteclose}\ \ \ \ {\isacharparenleft}{\isachardoublequoteopen}{\isacharunderscore}{\isachardoublequoteclose}\ {\isadigit{5}}{\isacharparenright}% +\ \ Trueprop\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}o\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ prop{\isaliteral{22}{\isachardoublequoteclose}}\ \ \ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5F}{\isacharunderscore}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{5}}{\isaliteral{29}{\isacharparenright}}% \begin{isamarkuptext}% \noindent Note that the object-logic judgement is implicit in the - syntax: writing \isa{A} produces \isa{{\isachardoublequote}Trueprop\ A{\isachardoublequote}} internally. + syntax: writing \isa{A} produces \isa{{\isaliteral{22}{\isachardoublequote}}Trueprop\ A{\isaliteral{22}{\isachardoublequote}}} internally. From the Pure perspective this means ``\isa{A} is derivable in the object-logic''.% \end{isamarkuptext}% @@ -55,16 +55,16 @@ reflexivity as introduction, and substitution as elimination principle. Note that the latter is particularly convenient in a framework like Isabelle, because syntactic congruences are - implicitly produced by unification of \isa{{\isachardoublequote}B\ x{\isachardoublequote}} against + implicitly produced by unification of \isa{{\isaliteral{22}{\isachardoublequote}}B\ x{\isaliteral{22}{\isachardoublequote}}} against expressions containing occurrences of \isa{x}.% \end{isamarkuptext}% \isamarkuptrue% \isacommand{axiomatization}\isamarkupfalse% \isanewline -\ \ equal\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}i\ {\isasymRightarrow}\ i\ {\isasymRightarrow}\ o{\isachardoublequoteclose}\ \ {\isacharparenleft}\isakeyword{infix}\ {\isachardoublequoteopen}{\isacharequal}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}\isanewline +\ \ equal\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}i\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ i\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ o{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{28}{\isacharparenleft}}\isakeyword{infix}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{3D}{\isacharequal}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{5}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline \isakeyword{where}\isanewline -\ \ refl\ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline -\ \ subst\ {\isacharbrackleft}elim{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isacharequal}\ y\ {\isasymLongrightarrow}\ B\ x\ {\isasymLongrightarrow}\ B\ y{\isachardoublequoteclose}% +\ \ refl\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\isanewline +\ \ subst\ {\isaliteral{5B}{\isacharbrackleft}}elim{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{3D}{\isacharequal}}\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ y{\isaliteral{22}{\isachardoublequoteclose}}% \begin{isamarkuptext}% \noindent Substitution is very powerful, but also hard to control in full generality. We derive some common symmetry~/ transitivity @@ -72,9 +72,9 @@ \end{isamarkuptext}% \isamarkuptrue% \isacommand{theorem}\isamarkupfalse% -\ sym\ {\isacharbrackleft}sym{\isacharbrackright}{\isacharcolon}\isanewline -\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}x\ {\isacharequal}\ y{\isachardoublequoteclose}\isanewline -\ \ \isakeyword{shows}\ {\isachardoublequoteopen}y\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline +\ sym\ {\isaliteral{5B}{\isacharbrackleft}}sym{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\isanewline +\ \ \isakeyword{assumes}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\ \ \isakeyword{shows}\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline % \isadelimproof % @@ -82,13 +82,13 @@ % \isatagproof \isacommand{proof}\isamarkupfalse% -\ {\isacharminus}\isanewline +\ {\isaliteral{2D}{\isacharminus}}\isanewline \ \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}x\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% \isanewline \ \ \isacommand{with}\isamarkupfalse% -\ {\isacharbackquoteopen}x\ {\isacharequal}\ y{\isacharbackquoteclose}\ \isacommand{show}\isamarkupfalse% -\ {\isachardoublequoteopen}y\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ {\isaliteral{60}{\isacharbackquoteopen}}x\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{show}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% \isanewline \isacommand{qed}\isamarkupfalse% % @@ -101,9 +101,9 @@ \endisadelimproof \isanewline \isacommand{theorem}\isamarkupfalse% -\ forw{\isacharunderscore}subst\ {\isacharbrackleft}trans{\isacharbrackright}{\isacharcolon}\isanewline -\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}y\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}B\ x{\isachardoublequoteclose}\isanewline -\ \ \isakeyword{shows}\ {\isachardoublequoteopen}B\ y{\isachardoublequoteclose}\isanewline +\ forw{\isaliteral{5F}{\isacharunderscore}}subst\ {\isaliteral{5B}{\isacharbrackleft}}trans{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\isanewline +\ \ \isakeyword{assumes}\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}B\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\ \ \isakeyword{shows}\ {\isaliteral{22}{\isachardoublequoteopen}}B\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline % \isadelimproof % @@ -111,14 +111,14 @@ % \isatagproof \isacommand{proof}\isamarkupfalse% -\ {\isacharminus}\isanewline +\ {\isaliteral{2D}{\isacharminus}}\isanewline \ \ \isacommand{from}\isamarkupfalse% -\ {\isacharbackquoteopen}y\ {\isacharequal}\ x{\isacharbackquoteclose}\ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}x\ {\isacharequal}\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ {\isaliteral{60}{\isacharbackquoteopen}}y\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{have}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% \isanewline \ \ \isacommand{from}\isamarkupfalse% -\ this\ \isakeyword{and}\ {\isacharbackquoteopen}B\ x{\isacharbackquoteclose}\ \isacommand{show}\isamarkupfalse% -\ {\isachardoublequoteopen}B\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ this\ \isakeyword{and}\ {\isaliteral{60}{\isacharbackquoteopen}}B\ x{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{show}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}B\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% \isanewline \isacommand{qed}\isamarkupfalse% % @@ -131,9 +131,9 @@ \endisadelimproof \isanewline \isacommand{theorem}\isamarkupfalse% -\ back{\isacharunderscore}subst\ {\isacharbrackleft}trans{\isacharbrackright}{\isacharcolon}\isanewline -\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}B\ x{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}x\ {\isacharequal}\ y{\isachardoublequoteclose}\isanewline -\ \ \isakeyword{shows}\ {\isachardoublequoteopen}B\ y{\isachardoublequoteclose}\isanewline +\ back{\isaliteral{5F}{\isacharunderscore}}subst\ {\isaliteral{5B}{\isacharbrackleft}}trans{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\isanewline +\ \ \isakeyword{assumes}\ {\isaliteral{22}{\isachardoublequoteopen}}B\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\ \ \isakeyword{shows}\ {\isaliteral{22}{\isachardoublequoteopen}}B\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline % \isadelimproof % @@ -141,11 +141,11 @@ % \isatagproof \isacommand{proof}\isamarkupfalse% -\ {\isacharminus}\isanewline +\ {\isaliteral{2D}{\isacharminus}}\isanewline \ \ \isacommand{from}\isamarkupfalse% -\ {\isacharbackquoteopen}x\ {\isacharequal}\ y{\isacharbackquoteclose}\ \isakeyword{and}\ {\isacharbackquoteopen}B\ x{\isacharbackquoteclose}\isanewline +\ {\isaliteral{60}{\isacharbackquoteopen}}x\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{60}{\isacharbackquoteclose}}\ \isakeyword{and}\ {\isaliteral{60}{\isacharbackquoteopen}}B\ x{\isaliteral{60}{\isacharbackquoteclose}}\isanewline \ \ \isacommand{show}\isamarkupfalse% -\ {\isachardoublequoteopen}B\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}B\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% \isanewline \isacommand{qed}\isamarkupfalse% % @@ -158,9 +158,9 @@ \endisadelimproof \isanewline \isacommand{theorem}\isamarkupfalse% -\ trans\ {\isacharbrackleft}trans{\isacharbrackright}{\isacharcolon}\isanewline -\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}x\ {\isacharequal}\ y{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}y\ {\isacharequal}\ z{\isachardoublequoteclose}\isanewline -\ \ \isakeyword{shows}\ {\isachardoublequoteopen}x\ {\isacharequal}\ z{\isachardoublequoteclose}\isanewline +\ trans\ {\isaliteral{5B}{\isacharbrackleft}}trans{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\isanewline +\ \ \isakeyword{assumes}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}y\ {\isaliteral{3D}{\isacharequal}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\ \ \isakeyword{shows}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{3D}{\isacharequal}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\isanewline % \isadelimproof % @@ -168,11 +168,11 @@ % \isatagproof \isacommand{proof}\isamarkupfalse% -\ {\isacharminus}\isanewline +\ {\isaliteral{2D}{\isacharminus}}\isanewline \ \ \isacommand{from}\isamarkupfalse% -\ {\isacharbackquoteopen}y\ {\isacharequal}\ z{\isacharbackquoteclose}\ \isakeyword{and}\ {\isacharbackquoteopen}x\ {\isacharequal}\ y{\isacharbackquoteclose}\isanewline +\ {\isaliteral{60}{\isacharbackquoteopen}}y\ {\isaliteral{3D}{\isacharequal}}\ z{\isaliteral{60}{\isacharbackquoteclose}}\ \isakeyword{and}\ {\isaliteral{60}{\isacharbackquoteopen}}x\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{60}{\isacharbackquoteclose}}\isanewline \ \ \isacommand{show}\isamarkupfalse% -\ {\isachardoublequoteopen}x\ {\isacharequal}\ z{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{3D}{\isacharequal}}\ z{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% \isanewline \isacommand{qed}\isamarkupfalse% % @@ -195,17 +195,17 @@ \end{isamarkuptext}% \isamarkuptrue% \isacommand{locale}\isamarkupfalse% -\ group\ {\isacharequal}\isanewline -\ \ \isakeyword{fixes}\ prod\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}i\ {\isasymRightarrow}\ i\ {\isasymRightarrow}\ i{\isachardoublequoteclose}\ \ {\isacharparenleft}\isakeyword{infix}\ {\isachardoublequoteopen}{\isasymcirc}{\isachardoublequoteclose}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline -\ \ \ \ \isakeyword{and}\ inv\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}i\ {\isasymRightarrow}\ i{\isachardoublequoteclose}\ \ {\isacharparenleft}{\isachardoublequoteopen}{\isacharparenleft}{\isacharunderscore}{\isasyminverse}{\isacharparenright}{\isachardoublequoteclose}\ {\isacharbrackleft}{\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharbrackright}\ {\isadigit{9}}{\isadigit{9}}{\isadigit{9}}{\isacharparenright}\isanewline -\ \ \ \ \isakeyword{and}\ unit\ {\isacharcolon}{\isacharcolon}\ i\ \ {\isacharparenleft}{\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}{\isacharparenright}\isanewline -\ \ \isakeyword{assumes}\ assoc{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymcirc}\ y{\isacharparenright}\ {\isasymcirc}\ z\ {\isacharequal}\ x\ {\isasymcirc}\ {\isacharparenleft}y\ {\isasymcirc}\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline -\ \ \ \ \isakeyword{and}\ left{\isacharunderscore}unit{\isacharcolon}\ \ {\isachardoublequoteopen}{\isadigit{1}}\ {\isasymcirc}\ x\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline -\ \ \ \ \isakeyword{and}\ left{\isacharunderscore}inv{\isacharcolon}\ {\isachardoublequoteopen}x{\isasyminverse}\ {\isasymcirc}\ x\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline +\ group\ {\isaliteral{3D}{\isacharequal}}\isanewline +\ \ \isakeyword{fixes}\ prod\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}i\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ i\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ i{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{28}{\isacharparenleft}}\isakeyword{infix}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C636972633E}{\isasymcirc}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{7}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline +\ \ \ \ \isakeyword{and}\ inv\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}i\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ i{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5F}{\isacharunderscore}}{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isaliteral{5D}{\isacharbrackright}}\ {\isadigit{9}}{\isadigit{9}}{\isadigit{9}}{\isaliteral{29}{\isacharparenright}}\isanewline +\ \ \ \ \isakeyword{and}\ unit\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ i\ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{22}{\isachardoublequoteopen}}{\isadigit{1}}{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}\isanewline +\ \ \isakeyword{assumes}\ assoc{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ z\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\ \ \ \ \isakeyword{and}\ left{\isaliteral{5F}{\isacharunderscore}}unit{\isaliteral{3A}{\isacharcolon}}\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isadigit{1}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ x\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\ \ \ \ \isakeyword{and}\ left{\isaliteral{5F}{\isacharunderscore}}inv{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline \isakeyword{begin}\isanewline \isanewline \isacommand{theorem}\isamarkupfalse% -\ right{\isacharunderscore}inv{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymcirc}\ x{\isasyminverse}\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline +\ right{\isaliteral{5F}{\isacharunderscore}}inv{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ x{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline % \isadelimproof % @@ -213,41 +213,41 @@ % \isatagproof \isacommand{proof}\isamarkupfalse% -\ {\isacharminus}\isanewline +\ {\isaliteral{2D}{\isacharminus}}\isanewline \ \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}x\ {\isasymcirc}\ x{\isasyminverse}\ {\isacharequal}\ {\isadigit{1}}\ {\isasymcirc}\ {\isacharparenleft}x\ {\isasymcirc}\ x{\isasyminverse}{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% -\ {\isacharparenleft}rule\ left{\isacharunderscore}unit\ {\isacharbrackleft}symmetric{\isacharbrackright}{\isacharparenright}\isanewline +\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ x{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ x{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse% +\ {\isaliteral{28}{\isacharparenleft}}rule\ left{\isaliteral{5F}{\isacharunderscore}}unit\ {\isaliteral{5B}{\isacharbrackleft}}symmetric{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\isanewline \ \ \isacommand{also}\isamarkupfalse% \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{1}}\ {\isasymcirc}\ x{\isacharparenright}\ {\isasymcirc}\ x{\isasyminverse}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% -\ {\isacharparenleft}rule\ assoc\ {\isacharbrackleft}symmetric{\isacharbrackright}{\isacharparenright}\isanewline +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{1}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ x{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse% +\ {\isaliteral{28}{\isacharparenleft}}rule\ assoc\ {\isaliteral{5B}{\isacharbrackleft}}symmetric{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\isanewline \ \ \isacommand{also}\isamarkupfalse% \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}{\isadigit{1}}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminverse}{\isacharparenright}{\isasyminverse}\ {\isasymcirc}\ x{\isasyminverse}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% -\ {\isacharparenleft}rule\ left{\isacharunderscore}inv\ {\isacharbrackleft}symmetric{\isacharbrackright}{\isacharparenright}\isanewline +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isadigit{1}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ x{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse% +\ {\isaliteral{28}{\isacharparenleft}}rule\ left{\isaliteral{5F}{\isacharunderscore}}inv\ {\isaliteral{5B}{\isacharbrackleft}}symmetric{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\isanewline \ \ \isacommand{also}\isamarkupfalse% \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymcirc}\ x\ {\isacharequal}\ {\isacharparenleft}x{\isasyminverse}{\isacharparenright}{\isasyminverse}\ {\isasymcirc}\ {\isacharparenleft}x{\isasyminverse}\ {\isasymcirc}\ x{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% -\ {\isacharparenleft}rule\ assoc{\isacharparenright}\isanewline +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse% +\ {\isaliteral{28}{\isacharparenleft}}rule\ assoc{\isaliteral{29}{\isacharparenright}}\isanewline \ \ \isacommand{also}\isamarkupfalse% \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}x{\isasyminverse}\ {\isasymcirc}\ x\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% -\ {\isacharparenleft}rule\ left{\isacharunderscore}inv{\isacharparenright}\isanewline +\ {\isaliteral{22}{\isachardoublequoteopen}}x{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse% +\ {\isaliteral{28}{\isacharparenleft}}rule\ left{\isaliteral{5F}{\isacharunderscore}}inv{\isaliteral{29}{\isacharparenright}}\isanewline \ \ \isacommand{also}\isamarkupfalse% \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}{\isacharparenleft}{\isacharparenleft}x{\isasyminverse}{\isacharparenright}{\isasyminverse}\ {\isasymcirc}\ {\isasymdots}{\isacharparenright}\ {\isasymcirc}\ x{\isasyminverse}\ {\isacharequal}\ {\isacharparenleft}x{\isasyminverse}{\isacharparenright}{\isasyminverse}\ {\isasymcirc}\ {\isacharparenleft}{\isadigit{1}}\ {\isasymcirc}\ x{\isasyminverse}{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% -\ {\isacharparenleft}rule\ assoc{\isacharparenright}\isanewline +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ x{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{1}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ x{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse% +\ {\isaliteral{28}{\isacharparenleft}}rule\ assoc{\isaliteral{29}{\isacharparenright}}\isanewline \ \ \isacommand{also}\isamarkupfalse% \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}{\isadigit{1}}\ {\isasymcirc}\ x{\isasyminverse}\ {\isacharequal}\ x{\isasyminverse}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% -\ {\isacharparenleft}rule\ left{\isacharunderscore}unit{\isacharparenright}\isanewline +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isadigit{1}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ x{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse% +\ {\isaliteral{28}{\isacharparenleft}}rule\ left{\isaliteral{5F}{\isacharunderscore}}unit{\isaliteral{29}{\isacharparenright}}\isanewline \ \ \isacommand{also}\isamarkupfalse% \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}{\isacharparenleft}x{\isasyminverse}{\isacharparenright}{\isasyminverse}\ {\isasymcirc}\ {\isasymdots}\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% -\ {\isacharparenleft}rule\ left{\isacharunderscore}inv{\isacharparenright}\isanewline +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse% +\ {\isaliteral{28}{\isacharparenleft}}rule\ left{\isaliteral{5F}{\isacharunderscore}}inv{\isaliteral{29}{\isacharparenright}}\isanewline \ \ \isacommand{finally}\isamarkupfalse% \ \isacommand{show}\isamarkupfalse% -\ {\isachardoublequoteopen}x\ {\isasymcirc}\ x{\isasyminverse}\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isacommand{{\isachardot}}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ x{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% \isanewline \isacommand{qed}\isamarkupfalse% % @@ -260,7 +260,7 @@ \endisadelimproof \isanewline \isacommand{theorem}\isamarkupfalse% -\ right{\isacharunderscore}unit{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymcirc}\ {\isadigit{1}}\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline +\ right{\isaliteral{5F}{\isacharunderscore}}unit{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ {\isadigit{1}}\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline % \isadelimproof % @@ -268,25 +268,25 @@ % \isatagproof \isacommand{proof}\isamarkupfalse% -\ {\isacharminus}\isanewline +\ {\isaliteral{2D}{\isacharminus}}\isanewline \ \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}{\isadigit{1}}\ {\isacharequal}\ x{\isasyminverse}\ {\isasymcirc}\ x{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% -\ {\isacharparenleft}rule\ left{\isacharunderscore}inv\ {\isacharbrackleft}symmetric{\isacharbrackright}{\isacharparenright}\isanewline +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isadigit{1}}\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse% +\ {\isaliteral{28}{\isacharparenleft}}rule\ left{\isaliteral{5F}{\isacharunderscore}}inv\ {\isaliteral{5B}{\isacharbrackleft}}symmetric{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\isanewline \ \ \isacommand{also}\isamarkupfalse% \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}x\ {\isasymcirc}\ {\isasymdots}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymcirc}\ x{\isasyminverse}{\isacharparenright}\ {\isasymcirc}\ x{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% -\ {\isacharparenleft}rule\ assoc\ {\isacharbrackleft}symmetric{\isacharbrackright}{\isacharparenright}\isanewline +\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ x{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse% +\ {\isaliteral{28}{\isacharparenleft}}rule\ assoc\ {\isaliteral{5B}{\isacharbrackleft}}symmetric{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\isanewline \ \ \isacommand{also}\isamarkupfalse% \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}x\ {\isasymcirc}\ x{\isasyminverse}\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% -\ {\isacharparenleft}rule\ right{\isacharunderscore}inv{\isacharparenright}\isanewline +\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ x{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse% +\ {\isaliteral{28}{\isacharparenleft}}rule\ right{\isaliteral{5F}{\isacharunderscore}}inv{\isaliteral{29}{\isacharparenright}}\isanewline \ \ \isacommand{also}\isamarkupfalse% \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}{\isasymdots}\ {\isasymcirc}\ x\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse% -\ {\isacharparenleft}rule\ left{\isacharunderscore}unit{\isacharparenright}\isanewline +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ x\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse% +\ {\isaliteral{28}{\isacharparenleft}}rule\ left{\isaliteral{5F}{\isacharunderscore}}unit{\isaliteral{29}{\isacharparenright}}\isanewline \ \ \isacommand{finally}\isamarkupfalse% \ \isacommand{show}\isamarkupfalse% -\ {\isachardoublequoteopen}x\ {\isasymcirc}\ {\isadigit{1}}\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ {\isadigit{1}}\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% \isanewline \isacommand{qed}\isamarkupfalse% % @@ -300,12 +300,12 @@ \begin{isamarkuptext}% \noindent Reasoning from basic axioms is often tedious. Our proofs work by producing various instances of the given rules (potentially - the symmetric form) using the pattern ``\hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}~\isa{eq}~\hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\isa{{\isachardoublequote}{\isacharparenleft}rule\ r{\isacharparenright}{\isachardoublequote}}'' and composing the chain of + the symmetric form) using the pattern ``\hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}~\isa{eq}~\hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}rule\ r{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}'' and composing the chain of results via \hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}}/\hyperlink{command.finally}{\mbox{\isa{\isacommand{finally}}}}. These steps may involve any of the transitivity rules declared in \secref{sec:framework-ex-equal}, namely \isa{trans} in combining - the first two results in \isa{right{\isacharunderscore}inv} and in the final steps of - both proofs, \isa{forw{\isacharunderscore}subst} in the first combination of \isa{right{\isacharunderscore}unit}, and \isa{back{\isacharunderscore}subst} in all other calculational steps. + the first two results in \isa{right{\isaliteral{5F}{\isacharunderscore}}inv} and in the final steps of + both proofs, \isa{forw{\isaliteral{5F}{\isacharunderscore}}subst} in the first combination of \isa{right{\isaliteral{5F}{\isacharunderscore}}unit}, and \isa{back{\isaliteral{5F}{\isacharunderscore}}subst} in all other calculational steps. Occasional substitutions in calculations are adequate, but should not be over-emphasized. The other extreme is to compose a chain by @@ -320,27 +320,27 @@ % \isatagproof \ \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}x\ {\isasymcirc}\ {\isadigit{1}}\ {\isacharequal}\ x\ {\isasymcirc}\ {\isacharparenleft}x{\isasyminverse}\ {\isasymcirc}\ x{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{unfolding}\isamarkupfalse% -\ left{\isacharunderscore}inv\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ {\isadigit{1}}\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{unfolding}\isamarkupfalse% +\ left{\isaliteral{5F}{\isacharunderscore}}inv\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% \isanewline \ \ \isacommand{also}\isamarkupfalse% \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}\ {\isacharparenleft}x\ {\isasymcirc}\ x{\isasyminverse}{\isacharparenright}\ {\isasymcirc}\ x{\isachardoublequoteclose}\ \isacommand{unfolding}\isamarkupfalse% -\ assoc\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ x{\isaliteral{5C3C696E76657273653E}{\isasyminverse}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{unfolding}\isamarkupfalse% +\ assoc\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% \isanewline \ \ \isacommand{also}\isamarkupfalse% \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}\ {\isadigit{1}}\ {\isasymcirc}\ x{\isachardoublequoteclose}\ \isacommand{unfolding}\isamarkupfalse% -\ right{\isacharunderscore}inv\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{unfolding}\isamarkupfalse% +\ right{\isaliteral{5F}{\isacharunderscore}}inv\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% \isanewline \ \ \isacommand{also}\isamarkupfalse% \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}{\isasymdots}\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isacommand{unfolding}\isamarkupfalse% -\ left{\isacharunderscore}unit\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{unfolding}\isamarkupfalse% +\ left{\isaliteral{5F}{\isacharunderscore}}unit\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% \isanewline \ \ \isacommand{finally}\isamarkupfalse% \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}x\ {\isasymcirc}\ {\isadigit{1}}\ {\isacharequal}\ x{\isachardoublequoteclose}\ \isacommand{{\isachardot}}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ {\isadigit{1}}\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% % \endisatagproof {\isafoldproof}% @@ -354,7 +354,7 @@ definitions in order to normalize each equational problem. A more realistic object-logic would include proper setup for the Simplifier (\secref{sec:simplifier}), the main automated tool for equational - reasoning in Isabelle. Then ``\hyperlink{command.unfolding}{\mbox{\isa{\isacommand{unfolding}}}}~\isa{left{\isacharunderscore}inv}~\hyperlink{command.ddot}{\mbox{\isa{\isacommand{{\isachardot}{\isachardot}}}}}'' would become ``\hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\isa{{\isachardoublequote}{\isacharparenleft}simp\ only{\isacharcolon}\ left{\isacharunderscore}inv{\isacharparenright}{\isachardoublequote}}'' etc.% + reasoning in Isabelle. Then ``\hyperlink{command.unfolding}{\mbox{\isa{\isacommand{unfolding}}}}~\isa{left{\isaliteral{5F}{\isacharunderscore}}inv}~\hyperlink{command.ddot}{\mbox{\isa{\isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}}}}'' would become ``\hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}simp\ only{\isaliteral{3A}{\isacharcolon}}\ left{\isaliteral{5F}{\isacharunderscore}}inv{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}'' etc.% \end{isamarkuptext}% \isamarkuptrue% \isacommand{end}\isamarkupfalse% @@ -371,34 +371,34 @@ \isamarkuptrue% \isacommand{axiomatization}\isamarkupfalse% \isanewline -\ \ imp\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}o\ {\isasymRightarrow}\ o\ {\isasymRightarrow}\ o{\isachardoublequoteclose}\ \ {\isacharparenleft}\isakeyword{infixr}\ {\isachardoublequoteopen}{\isasymlongrightarrow}{\isachardoublequoteclose}\ {\isadigit{2}}{\isadigit{5}}{\isacharparenright}\ \isakeyword{where}\isanewline -\ \ impI\ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}A\ {\isasymLongrightarrow}\ B{\isacharparenright}\ {\isasymLongrightarrow}\ A\ {\isasymlongrightarrow}\ B{\isachardoublequoteclose}\ \isakeyword{and}\isanewline -\ \ impD\ {\isacharbrackleft}dest{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}A\ {\isasymlongrightarrow}\ B{\isacharparenright}\ {\isasymLongrightarrow}\ A\ {\isasymLongrightarrow}\ B{\isachardoublequoteclose}\isanewline +\ \ imp\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}o\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ o\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ o{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{28}{\isacharparenleft}}\isakeyword{infixr}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{2}}{\isadigit{5}}{\isaliteral{29}{\isacharparenright}}\ \isakeyword{where}\isanewline +\ \ impI\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\isanewline +\ \ impD\ {\isaliteral{5B}{\isacharbrackleft}}dest{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ B{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\isanewline \isanewline \isacommand{axiomatization}\isamarkupfalse% \isanewline -\ \ disj\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}o\ {\isasymRightarrow}\ o\ {\isasymRightarrow}\ o{\isachardoublequoteclose}\ \ {\isacharparenleft}\isakeyword{infixr}\ {\isachardoublequoteopen}{\isasymor}{\isachardoublequoteclose}\ {\isadigit{3}}{\isadigit{0}}{\isacharparenright}\ \isakeyword{where}\isanewline -\ \ disjI\isactrlisub {\isadigit{1}}\ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}A\ {\isasymLongrightarrow}\ A\ {\isasymor}\ B{\isachardoublequoteclose}\ \isakeyword{and}\isanewline -\ \ disjI\isactrlisub {\isadigit{2}}\ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}B\ {\isasymLongrightarrow}\ A\ {\isasymor}\ B{\isachardoublequoteclose}\ \isakeyword{and}\isanewline -\ \ disjE\ {\isacharbrackleft}elim{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}A\ {\isasymor}\ B\ {\isasymLongrightarrow}\ {\isacharparenleft}A\ {\isasymLongrightarrow}\ C{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}B\ {\isasymLongrightarrow}\ C{\isacharparenright}\ {\isasymLongrightarrow}\ C{\isachardoublequoteclose}\isanewline +\ \ disj\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}o\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ o\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ o{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{28}{\isacharparenleft}}\isakeyword{infixr}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6F723E}{\isasymor}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{3}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ \isakeyword{where}\isanewline +\ \ disjI\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A\ {\isaliteral{5C3C6F723E}{\isasymor}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\isanewline +\ \ disjI\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A\ {\isaliteral{5C3C6F723E}{\isasymor}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\isanewline +\ \ disjE\ {\isaliteral{5B}{\isacharbrackleft}}elim{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C6F723E}{\isasymor}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C{\isaliteral{22}{\isachardoublequoteclose}}\isanewline \isanewline \isacommand{axiomatization}\isamarkupfalse% \isanewline -\ \ conj\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}o\ {\isasymRightarrow}\ o\ {\isasymRightarrow}\ o{\isachardoublequoteclose}\ \ {\isacharparenleft}\isakeyword{infixr}\ {\isachardoublequoteopen}{\isasymand}{\isachardoublequoteclose}\ {\isadigit{3}}{\isadigit{5}}{\isacharparenright}\ \isakeyword{where}\isanewline -\ \ conjI\ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}A\ {\isasymLongrightarrow}\ B\ {\isasymLongrightarrow}\ A\ {\isasymand}\ B{\isachardoublequoteclose}\ \isakeyword{and}\isanewline -\ \ conjD\isactrlisub {\isadigit{1}}{\isacharcolon}\ {\isachardoublequoteopen}A\ {\isasymand}\ B\ {\isasymLongrightarrow}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline -\ \ conjD\isactrlisub {\isadigit{2}}{\isacharcolon}\ {\isachardoublequoteopen}A\ {\isasymand}\ B\ {\isasymLongrightarrow}\ B{\isachardoublequoteclose}% +\ \ conj\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}o\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ o\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ o{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{28}{\isacharparenleft}}\isakeyword{infixr}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C616E643E}{\isasymand}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{3}}{\isadigit{5}}{\isaliteral{29}{\isacharparenright}}\ \isakeyword{where}\isanewline +\ \ conjI\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\isanewline +\ \ conjD\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\isanewline +\ \ conjD\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B{\isaliteral{22}{\isachardoublequoteclose}}% \begin{isamarkuptext}% \noindent The conjunctive destructions have the disadvantage that - decomposing \isa{{\isachardoublequote}A\ {\isasymand}\ B{\isachardoublequote}} involves an immediate decision which + decomposing \isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B{\isaliteral{22}{\isachardoublequote}}} involves an immediate decision which component should be projected. The more convenient simultaneous - elimination \isa{{\isachardoublequote}A\ {\isasymand}\ B\ {\isasymLongrightarrow}\ {\isacharparenleft}A\ {\isasymLongrightarrow}\ B\ {\isasymLongrightarrow}\ C{\isacharparenright}\ {\isasymLongrightarrow}\ C{\isachardoublequote}} can be derived as + elimination \isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C{\isaliteral{22}{\isachardoublequote}}} can be derived as follows:% \end{isamarkuptext}% \isamarkuptrue% \isacommand{theorem}\isamarkupfalse% -\ conjE\ {\isacharbrackleft}elim{\isacharbrackright}{\isacharcolon}\isanewline -\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}A\ {\isasymand}\ B{\isachardoublequoteclose}\isanewline +\ conjE\ {\isaliteral{5B}{\isacharbrackleft}}elim{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\isanewline +\ \ \isakeyword{assumes}\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\isanewline \ \ \isakeyword{obtains}\ A\ \isakeyword{and}\ B\isanewline % \isadelimproof @@ -409,13 +409,13 @@ \isacommand{proof}\isamarkupfalse% \isanewline \ \ \isacommand{from}\isamarkupfalse% -\ {\isacharbackquoteopen}A\ {\isasymand}\ B{\isacharbackquoteclose}\ \isacommand{show}\isamarkupfalse% +\ {\isaliteral{60}{\isacharbackquoteopen}}A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{show}\isamarkupfalse% \ A\ \isacommand{by}\isamarkupfalse% -\ {\isacharparenleft}rule\ conjD\isactrlisub {\isadigit{1}}{\isacharparenright}\isanewline +\ {\isaliteral{28}{\isacharparenleft}}rule\ conjD\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\isanewline \ \ \isacommand{from}\isamarkupfalse% -\ {\isacharbackquoteopen}A\ {\isasymand}\ B{\isacharbackquoteclose}\ \isacommand{show}\isamarkupfalse% +\ {\isaliteral{60}{\isacharbackquoteopen}}A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{show}\isamarkupfalse% \ B\ \isacommand{by}\isamarkupfalse% -\ {\isacharparenleft}rule\ conjD\isactrlisub {\isadigit{2}}{\isacharparenright}\isanewline +\ {\isaliteral{28}{\isacharparenleft}}rule\ conjD\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\isanewline \isacommand{qed}\isamarkupfalse% % \endisatagproof @@ -437,14 +437,14 @@ % \isatagproof \ \ \isacommand{assume}\isamarkupfalse% -\ {\isachardoublequoteopen}A\ {\isasymand}\ B{\isachardoublequoteclose}\isanewline +\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\isanewline \ \ \isacommand{then}\isamarkupfalse% \ \isacommand{obtain}\isamarkupfalse% -\ B\ \isakeyword{and}\ A\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ B\ \isakeyword{and}\ A\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% \isanewline \ \ \isacommand{then}\isamarkupfalse% \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}B\ {\isasymand}\ A{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}B\ {\isaliteral{5C3C616E643E}{\isasymand}}\ A{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% % \endisatagproof {\isafoldproof}% @@ -455,7 +455,7 @@ % \begin{isamarkuptext}% \noindent Note that the analogous elimination rule for disjunction - ``\isa{{\isachardoublequote}{\isasymASSUMES}\ A\ {\isasymor}\ B\ {\isasymOBTAINS}\ A\ {\isasymBBAR}\ B{\isachardoublequote}}'' coincides with + ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C415353554D45533E}{\isasymASSUMES}}\ A\ {\isaliteral{5C3C6F723E}{\isasymor}}\ B\ {\isaliteral{5C3C4F425441494E533E}{\isasymOBTAINS}}\ A\ {\isaliteral{5C3C424241523E}{\isasymBBAR}}\ B{\isaliteral{22}{\isachardoublequote}}}'' coincides with the original axiomatization of \isa{disjE}. \medskip We continue propositional logic by introducing absurdity @@ -465,16 +465,16 @@ \isamarkuptrue% \isacommand{axiomatization}\isamarkupfalse% \isanewline -\ \ false\ {\isacharcolon}{\isacharcolon}\ o\ \ {\isacharparenleft}{\isachardoublequoteopen}{\isasymbottom}{\isachardoublequoteclose}{\isacharparenright}\ \isakeyword{where}\isanewline -\ \ falseE\ {\isacharbrackleft}elim{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymbottom}\ {\isasymLongrightarrow}\ A{\isachardoublequoteclose}\isanewline +\ \ false\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ o\ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C626F74746F6D3E}{\isasymbottom}}{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}\ \isakeyword{where}\isanewline +\ \ falseE\ {\isaliteral{5B}{\isacharbrackleft}}elim{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C626F74746F6D3E}{\isasymbottom}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A{\isaliteral{22}{\isachardoublequoteclose}}\isanewline \isanewline \isacommand{definition}\isamarkupfalse% \isanewline -\ \ true\ {\isacharcolon}{\isacharcolon}\ o\ \ {\isacharparenleft}{\isachardoublequoteopen}{\isasymtop}{\isachardoublequoteclose}{\isacharparenright}\ \isakeyword{where}\isanewline -\ \ {\isachardoublequoteopen}{\isasymtop}\ {\isasymequiv}\ {\isasymbottom}\ {\isasymlongrightarrow}\ {\isasymbottom}{\isachardoublequoteclose}\isanewline +\ \ true\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ o\ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C746F703E}{\isasymtop}}{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}\ \isakeyword{where}\isanewline +\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C746F703E}{\isasymtop}}\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ {\isaliteral{5C3C626F74746F6D3E}{\isasymbottom}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ {\isaliteral{5C3C626F74746F6D3E}{\isasymbottom}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline \isanewline \isacommand{theorem}\isamarkupfalse% -\ trueI\ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\ {\isasymtop}\isanewline +\ trueI\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C746F703E}{\isasymtop}}\isanewline % \isadelimproof \ \ % @@ -482,7 +482,7 @@ % \isatagproof \isacommand{unfolding}\isamarkupfalse% -\ true{\isacharunderscore}def\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ true{\isaliteral{5F}{\isacharunderscore}}def\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% % \endisatagproof {\isafoldproof}% @@ -498,13 +498,13 @@ \isamarkuptrue% \isacommand{definition}\isamarkupfalse% \isanewline -\ \ not\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}o\ {\isasymRightarrow}\ o{\isachardoublequoteclose}\ \ {\isacharparenleft}{\isachardoublequoteopen}{\isasymnot}\ {\isacharunderscore}{\isachardoublequoteclose}\ {\isacharbrackleft}{\isadigit{4}}{\isadigit{0}}{\isacharbrackright}\ {\isadigit{4}}{\isadigit{0}}{\isacharparenright}\ \isakeyword{where}\isanewline -\ \ {\isachardoublequoteopen}{\isasymnot}\ A\ {\isasymequiv}\ A\ {\isasymlongrightarrow}\ {\isasymbottom}{\isachardoublequoteclose}\isanewline +\ \ not\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}o\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ o{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ {\isaliteral{5F}{\isacharunderscore}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isadigit{4}}{\isadigit{0}}{\isaliteral{5D}{\isacharbrackright}}\ {\isadigit{4}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ \isakeyword{where}\isanewline +\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ A\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ A\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ {\isaliteral{5C3C626F74746F6D3E}{\isasymbottom}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline \isanewline \isacommand{theorem}\isamarkupfalse% -\ notI\ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\isanewline -\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}A\ {\isasymLongrightarrow}\ {\isasymbottom}{\isachardoublequoteclose}\isanewline -\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymnot}\ A{\isachardoublequoteclose}\isanewline +\ notI\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\isanewline +\ \ \isakeyword{assumes}\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C626F74746F6D3E}{\isasymbottom}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\ \ \isakeyword{shows}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ A{\isaliteral{22}{\isachardoublequoteclose}}\isanewline % \isadelimproof % @@ -512,15 +512,15 @@ % \isatagproof \isacommand{unfolding}\isamarkupfalse% -\ not{\isacharunderscore}def\isanewline +\ not{\isaliteral{5F}{\isacharunderscore}}def\isanewline \isacommand{proof}\isamarkupfalse% \isanewline \ \ \isacommand{assume}\isamarkupfalse% \ A\isanewline \ \ \isacommand{then}\isamarkupfalse% \ \isacommand{show}\isamarkupfalse% -\ {\isasymbottom}\ \isacommand{by}\isamarkupfalse% -\ {\isacharparenleft}rule\ {\isacharbackquoteopen}A\ {\isasymLongrightarrow}\ {\isasymbottom}{\isacharbackquoteclose}{\isacharparenright}\isanewline +\ {\isaliteral{5C3C626F74746F6D3E}{\isasymbottom}}\ \isacommand{by}\isamarkupfalse% +\ {\isaliteral{28}{\isacharparenleft}}rule\ {\isaliteral{60}{\isacharbackquoteopen}}A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C626F74746F6D3E}{\isasymbottom}}{\isaliteral{60}{\isacharbackquoteclose}}{\isaliteral{29}{\isacharparenright}}\isanewline \isacommand{qed}\isamarkupfalse% % \endisatagproof @@ -532,8 +532,8 @@ \endisadelimproof \isanewline \isacommand{theorem}\isamarkupfalse% -\ notE\ {\isacharbrackleft}elim{\isacharbrackright}{\isacharcolon}\isanewline -\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymnot}\ A{\isachardoublequoteclose}\ \isakeyword{and}\ A\isanewline +\ notE\ {\isaliteral{5B}{\isacharbrackleft}}elim{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\isanewline +\ \ \isakeyword{assumes}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ A{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ A\isanewline \ \ \isakeyword{shows}\ B\isanewline % \isadelimproof @@ -542,19 +542,19 @@ % \isatagproof \isacommand{proof}\isamarkupfalse% -\ {\isacharminus}\isanewline +\ {\isaliteral{2D}{\isacharminus}}\isanewline \ \ \isacommand{from}\isamarkupfalse% -\ {\isacharbackquoteopen}{\isasymnot}\ A{\isacharbackquoteclose}\ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}A\ {\isasymlongrightarrow}\ {\isasymbottom}{\isachardoublequoteclose}\ \isacommand{unfolding}\isamarkupfalse% -\ not{\isacharunderscore}def\ \isacommand{{\isachardot}}\isamarkupfalse% +\ {\isaliteral{60}{\isacharbackquoteopen}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ A{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{have}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ {\isaliteral{5C3C626F74746F6D3E}{\isasymbottom}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{unfolding}\isamarkupfalse% +\ not{\isaliteral{5F}{\isacharunderscore}}def\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% \isanewline \ \ \isacommand{from}\isamarkupfalse% -\ {\isacharbackquoteopen}A\ {\isasymlongrightarrow}\ {\isasymbottom}{\isacharbackquoteclose}\ \isakeyword{and}\ {\isacharbackquoteopen}A{\isacharbackquoteclose}\ \isacommand{have}\isamarkupfalse% -\ {\isasymbottom}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ {\isaliteral{60}{\isacharbackquoteopen}}A\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ {\isaliteral{5C3C626F74746F6D3E}{\isasymbottom}}{\isaliteral{60}{\isacharbackquoteclose}}\ \isakeyword{and}\ {\isaliteral{60}{\isacharbackquoteopen}}A{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{have}\isamarkupfalse% +\ {\isaliteral{5C3C626F74746F6D3E}{\isasymbottom}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% \isanewline \ \ \isacommand{then}\isamarkupfalse% \ \isacommand{show}\isamarkupfalse% -\ B\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ B\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% \isanewline \isacommand{qed}\isamarkupfalse% % @@ -577,13 +577,13 @@ \end{isamarkuptext}% \isamarkuptrue% \isacommand{locale}\isamarkupfalse% -\ classical\ {\isacharequal}\isanewline -\ \ \isakeyword{assumes}\ classical{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}{\isasymnot}\ C\ {\isasymLongrightarrow}\ C{\isacharparenright}\ {\isasymLongrightarrow}\ C{\isachardoublequoteclose}\isanewline +\ classical\ {\isaliteral{3D}{\isacharequal}}\isanewline +\ \ \isakeyword{assumes}\ classical{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ C\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C{\isaliteral{22}{\isachardoublequoteclose}}\isanewline \isakeyword{begin}\isanewline \isanewline \isacommand{theorem}\isamarkupfalse% -\ double{\isacharunderscore}negation{\isacharcolon}\isanewline -\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymnot}\ {\isasymnot}\ C{\isachardoublequoteclose}\isanewline +\ double{\isaliteral{5F}{\isacharunderscore}}negation{\isaliteral{3A}{\isacharcolon}}\isanewline +\ \ \isakeyword{assumes}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ C{\isaliteral{22}{\isachardoublequoteclose}}\isanewline \ \ \isakeyword{shows}\ C\isanewline % \isadelimproof @@ -592,12 +592,12 @@ % \isatagproof \isacommand{proof}\isamarkupfalse% -\ {\isacharparenleft}rule\ classical{\isacharparenright}\isanewline +\ {\isaliteral{28}{\isacharparenleft}}rule\ classical{\isaliteral{29}{\isacharparenright}}\isanewline \ \ \isacommand{assume}\isamarkupfalse% -\ {\isachardoublequoteopen}{\isasymnot}\ C{\isachardoublequoteclose}\isanewline +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ C{\isaliteral{22}{\isachardoublequoteclose}}\isanewline \ \ \isacommand{with}\isamarkupfalse% -\ {\isacharbackquoteopen}{\isasymnot}\ {\isasymnot}\ C{\isacharbackquoteclose}\ \isacommand{show}\isamarkupfalse% -\ C\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ {\isaliteral{60}{\isacharbackquoteopen}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ C{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{show}\isamarkupfalse% +\ C\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% \isanewline \isacommand{qed}\isamarkupfalse% % @@ -610,7 +610,7 @@ \endisadelimproof \isanewline \isacommand{theorem}\isamarkupfalse% -\ tertium{\isacharunderscore}non{\isacharunderscore}datur{\isacharcolon}\ {\isachardoublequoteopen}C\ {\isasymor}\ {\isasymnot}\ C{\isachardoublequoteclose}\isanewline +\ tertium{\isaliteral{5F}{\isacharunderscore}}non{\isaliteral{5F}{\isacharunderscore}}datur{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}C\ {\isaliteral{5C3C6F723E}{\isasymor}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ C{\isaliteral{22}{\isachardoublequoteclose}}\isanewline % \isadelimproof % @@ -618,35 +618,35 @@ % \isatagproof \isacommand{proof}\isamarkupfalse% -\ {\isacharparenleft}rule\ double{\isacharunderscore}negation{\isacharparenright}\isanewline +\ {\isaliteral{28}{\isacharparenleft}}rule\ double{\isaliteral{5F}{\isacharunderscore}}negation{\isaliteral{29}{\isacharparenright}}\isanewline \ \ \isacommand{show}\isamarkupfalse% -\ {\isachardoublequoteopen}{\isasymnot}\ {\isasymnot}\ {\isacharparenleft}C\ {\isasymor}\ {\isasymnot}\ C{\isacharparenright}{\isachardoublequoteclose}\isanewline +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ {\isaliteral{28}{\isacharparenleft}}C\ {\isaliteral{5C3C6F723E}{\isasymor}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ C{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline \ \ \isacommand{proof}\isamarkupfalse% \isanewline \ \ \ \ \isacommand{assume}\isamarkupfalse% -\ {\isachardoublequoteopen}{\isasymnot}\ {\isacharparenleft}C\ {\isasymor}\ {\isasymnot}\ C{\isacharparenright}{\isachardoublequoteclose}\isanewline +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ {\isaliteral{28}{\isacharparenleft}}C\ {\isaliteral{5C3C6F723E}{\isasymor}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ C{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline \ \ \ \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}{\isasymnot}\ C{\isachardoublequoteclose}\isanewline +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ C{\isaliteral{22}{\isachardoublequoteclose}}\isanewline \ \ \ \ \isacommand{proof}\isamarkupfalse% \isanewline \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse% \ C\ \isacommand{then}\isamarkupfalse% \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}C\ {\isasymor}\ {\isasymnot}\ C{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}C\ {\isaliteral{5C3C6F723E}{\isasymor}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ C{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% \isanewline \ \ \ \ \ \ \isacommand{with}\isamarkupfalse% -\ {\isacharbackquoteopen}{\isasymnot}\ {\isacharparenleft}C\ {\isasymor}\ {\isasymnot}\ C{\isacharparenright}{\isacharbackquoteclose}\ \isacommand{show}\isamarkupfalse% -\ {\isasymbottom}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ {\isaliteral{60}{\isacharbackquoteopen}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ {\isaliteral{28}{\isacharparenleft}}C\ {\isaliteral{5C3C6F723E}{\isasymor}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ C{\isaliteral{29}{\isacharparenright}}{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{show}\isamarkupfalse% +\ {\isaliteral{5C3C626F74746F6D3E}{\isasymbottom}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% \isanewline \ \ \ \ \isacommand{qed}\isamarkupfalse% \isanewline \ \ \ \ \isacommand{then}\isamarkupfalse% \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}C\ {\isasymor}\ {\isasymnot}\ C{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}C\ {\isaliteral{5C3C6F723E}{\isasymor}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ C{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% \isanewline \ \ \ \ \isacommand{with}\isamarkupfalse% -\ {\isacharbackquoteopen}{\isasymnot}\ {\isacharparenleft}C\ {\isasymor}\ {\isasymnot}\ C{\isacharparenright}{\isacharbackquoteclose}\ \isacommand{show}\isamarkupfalse% -\ {\isasymbottom}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ {\isaliteral{60}{\isacharbackquoteopen}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ {\isaliteral{28}{\isacharparenleft}}C\ {\isaliteral{5C3C6F723E}{\isasymor}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ C{\isaliteral{29}{\isacharparenright}}{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{show}\isamarkupfalse% +\ {\isaliteral{5C3C626F74746F6D3E}{\isasymbottom}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% \isanewline \ \ \isacommand{qed}\isamarkupfalse% \isanewline @@ -664,9 +664,9 @@ non-trivial propositional proofs in general. All three rules characterize classical logic independently, but the original rule is already the most convenient to use, because it leaves the conclusion - unchanged. Note that \isa{{\isachardoublequote}{\isacharparenleft}{\isasymnot}\ C\ {\isasymLongrightarrow}\ C{\isacharparenright}\ {\isasymLongrightarrow}\ C{\isachardoublequote}} fits again into our + unchanged. Note that \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ C\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C{\isaliteral{22}{\isachardoublequote}}} fits again into our format for eliminations, despite the additional twist that the - context refers to the main conclusion. So we may write \isa{classical} as the Isar statement ``\isa{{\isachardoublequote}{\isasymOBTAINS}\ {\isasymnot}\ thesis{\isachardoublequote}}''. + context refers to the main conclusion. So we may write \isa{classical} as the Isar statement ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C4F425441494E533E}{\isasymOBTAINS}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ thesis{\isaliteral{22}{\isachardoublequote}}}''. This also explains nicely how classical reasoning really works: whatever the main \isa{thesis} might be, we may always assume its negation!% @@ -682,31 +682,31 @@ Representing quantifiers is easy, thanks to the higher-order nature of the underlying framework. According to the well-known technique introduced by Church \cite{church40}, quantifiers are operators on - predicates, which are syntactically represented as \isa{{\isachardoublequote}{\isasymlambda}{\isachardoublequote}}-terms - of type \isa{{\isachardoublequote}i\ {\isasymRightarrow}\ o{\isachardoublequote}}. Binder notation turns \isa{{\isachardoublequote}All\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ B\ x{\isacharparenright}{\isachardoublequote}} into \isa{{\isachardoublequote}{\isasymforall}x{\isachardot}\ B\ x{\isachardoublequote}} etc.% + predicates, which are syntactically represented as \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}{\isaliteral{22}{\isachardoublequote}}}-terms + of type \isa{{\isaliteral{22}{\isachardoublequote}}i\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ o{\isaliteral{22}{\isachardoublequote}}}. Binder notation turns \isa{{\isaliteral{22}{\isachardoublequote}}All\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ B\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} into \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ B\ x{\isaliteral{22}{\isachardoublequote}}} etc.% \end{isamarkuptext}% \isamarkuptrue% \isacommand{axiomatization}\isamarkupfalse% \isanewline -\ \ All\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}i\ {\isasymRightarrow}\ o{\isacharparenright}\ {\isasymRightarrow}\ o{\isachardoublequoteclose}\ \ {\isacharparenleft}\isakeyword{binder}\ {\isachardoublequoteopen}{\isasymforall}{\isachardoublequoteclose}\ {\isadigit{1}}{\isadigit{0}}{\isacharparenright}\ \isakeyword{where}\isanewline -\ \ allI\ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}{\isasymAnd}x{\isachardot}\ B\ x{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymforall}x{\isachardot}\ B\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline -\ \ allD\ {\isacharbrackleft}dest{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}{\isasymforall}x{\isachardot}\ B\ x{\isacharparenright}\ {\isasymLongrightarrow}\ B\ a{\isachardoublequoteclose}\isanewline +\ \ All\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ o{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ o{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{28}{\isacharparenleft}}\isakeyword{binder}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{1}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ \isakeyword{where}\isanewline +\ \ allI\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ B\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ B\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\isanewline +\ \ allD\ {\isaliteral{5B}{\isacharbrackleft}}dest{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ B\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ a{\isaliteral{22}{\isachardoublequoteclose}}\isanewline \isanewline \isacommand{axiomatization}\isamarkupfalse% \isanewline -\ \ Ex\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}i\ {\isasymRightarrow}\ o{\isacharparenright}\ {\isasymRightarrow}\ o{\isachardoublequoteclose}\ \ {\isacharparenleft}\isakeyword{binder}\ {\isachardoublequoteopen}{\isasymexists}{\isachardoublequoteclose}\ {\isadigit{1}}{\isadigit{0}}{\isacharparenright}\ \isakeyword{where}\isanewline -\ \ exI\ {\isacharbrackleft}intro{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}B\ a\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymexists}x{\isachardot}\ B\ x{\isacharparenright}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline -\ \ exE\ {\isacharbrackleft}elim{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}{\isasymexists}x{\isachardot}\ B\ x{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isasymAnd}x{\isachardot}\ B\ x\ {\isasymLongrightarrow}\ C{\isacharparenright}\ {\isasymLongrightarrow}\ C{\isachardoublequoteclose}% +\ \ Ex\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ o{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ o{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{28}{\isacharparenleft}}\isakeyword{binder}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{1}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ \isakeyword{where}\isanewline +\ \ exI\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}B\ a\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ B\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\isanewline +\ \ exE\ {\isaliteral{5B}{\isacharbrackleft}}elim{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ B\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ B\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C{\isaliteral{22}{\isachardoublequoteclose}}% \begin{isamarkuptext}% -\noindent The statement of \isa{exE} corresponds to ``\isa{{\isachardoublequote}{\isasymASSUMES}\ {\isasymexists}x{\isachardot}\ B\ x\ {\isasymOBTAINS}\ x\ {\isasymWHERE}\ B\ x{\isachardoublequote}}'' in Isar. In the +\noindent The statement of \isa{exE} corresponds to ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C415353554D45533E}{\isasymASSUMES}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ B\ x\ {\isaliteral{5C3C4F425441494E533E}{\isasymOBTAINS}}\ x\ {\isaliteral{5C3C57484552453E}{\isasymWHERE}}\ B\ x{\isaliteral{22}{\isachardoublequote}}}'' in Isar. In the subsequent example we illustrate quantifier reasoning involving all four rules:% \end{isamarkuptext}% \isamarkuptrue% \isacommand{theorem}\isamarkupfalse% \isanewline -\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}{\isasymexists}x{\isachardot}\ {\isasymforall}y{\isachardot}\ R\ x\ y{\isachardoublequoteclose}\isanewline -\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymforall}y{\isachardot}\ {\isasymexists}x{\isachardot}\ R\ x\ y{\isachardoublequoteclose}\isanewline +\ \ \isakeyword{assumes}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}y{\isaliteral{2E}{\isachardot}}\ R\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline +\ \ \isakeyword{shows}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}y{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ R\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline % \isadelimproof % @@ -715,29 +715,29 @@ \isatagproof \isacommand{proof}\isamarkupfalse% \ \ \ \ % -\isamarkupcmt{\isa{{\isachardoublequote}{\isasymforall}{\isachardoublequote}} introduction% +\isamarkupcmt{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}{\isaliteral{22}{\isachardoublequote}}} introduction% } \isanewline \ \ \isacommand{obtain}\isamarkupfalse% -\ x\ \isakeyword{where}\ {\isachardoublequoteopen}{\isasymforall}y{\isachardot}\ R\ x\ y{\isachardoublequoteclose}\ \isacommand{using}\isamarkupfalse% -\ {\isacharbackquoteopen}{\isasymexists}x{\isachardot}\ {\isasymforall}y{\isachardot}\ R\ x\ y{\isacharbackquoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ x\ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}y{\isaliteral{2E}{\isachardot}}\ R\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{using}\isamarkupfalse% +\ {\isaliteral{60}{\isacharbackquoteopen}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}y{\isaliteral{2E}{\isachardot}}\ R\ x\ y{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% \ \ \ \ % -\isamarkupcmt{\isa{{\isachardoublequote}{\isasymexists}{\isachardoublequote}} elimination% +\isamarkupcmt{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}{\isaliteral{22}{\isachardoublequote}}} elimination% } \isanewline \ \ \isacommand{fix}\isamarkupfalse% \ y\ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}R\ x\ y{\isachardoublequoteclose}\ \isacommand{using}\isamarkupfalse% -\ {\isacharbackquoteopen}{\isasymforall}y{\isachardot}\ R\ x\ y{\isacharbackquoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}R\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{using}\isamarkupfalse% +\ {\isaliteral{60}{\isacharbackquoteopen}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}y{\isaliteral{2E}{\isachardot}}\ R\ x\ y{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% \ \ \ \ % -\isamarkupcmt{\isa{{\isachardoublequote}{\isasymforall}{\isachardoublequote}} destruction% +\isamarkupcmt{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}{\isaliteral{22}{\isachardoublequote}}} destruction% } \isanewline \ \ \isacommand{then}\isamarkupfalse% \ \isacommand{show}\isamarkupfalse% -\ {\isachardoublequoteopen}{\isasymexists}x{\isachardot}\ R\ x\ y{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ R\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% \ \ \ \ % -\isamarkupcmt{\isa{{\isachardoublequote}{\isasymexists}{\isachardoublequote}} introduction% +\isamarkupcmt{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}{\isaliteral{22}{\isachardoublequote}}} introduction% } \isanewline \isacommand{qed}\isamarkupfalse% @@ -761,27 +761,27 @@ \medskip \begin{tabular}{l} - \isa{{\isachardoublequote}impI{\isacharcolon}\ {\isasymASSUMES}\ A\ {\isasymLongrightarrow}\ B\ {\isasymSHOWS}\ A\ {\isasymlongrightarrow}\ B{\isachardoublequote}} \\ - \isa{{\isachardoublequote}impD{\isacharcolon}\ {\isasymASSUMES}\ A\ {\isasymlongrightarrow}\ B\ {\isasymAND}\ A\ {\isasymSHOWS}\ B{\isachardoublequote}} \\[1ex] + \isa{{\isaliteral{22}{\isachardoublequote}}impI{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C415353554D45533E}{\isasymASSUMES}}\ A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ {\isaliteral{5C3C53484F57533E}{\isasymSHOWS}}\ A\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ B{\isaliteral{22}{\isachardoublequote}}} \\ + \isa{{\isaliteral{22}{\isachardoublequote}}impD{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C415353554D45533E}{\isasymASSUMES}}\ A\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ B\ {\isaliteral{5C3C414E443E}{\isasymAND}}\ A\ {\isaliteral{5C3C53484F57533E}{\isasymSHOWS}}\ B{\isaliteral{22}{\isachardoublequote}}} \\[1ex] - \isa{{\isachardoublequote}disjI\isactrlisub {\isadigit{1}}{\isacharcolon}\ {\isasymASSUMES}\ A\ {\isasymSHOWS}\ A\ {\isasymor}\ B{\isachardoublequote}} \\ - \isa{{\isachardoublequote}disjI\isactrlisub {\isadigit{2}}{\isacharcolon}\ {\isasymASSUMES}\ B\ {\isasymSHOWS}\ A\ {\isasymor}\ B{\isachardoublequote}} \\ - \isa{{\isachardoublequote}disjE{\isacharcolon}\ {\isasymASSUMES}\ A\ {\isasymor}\ B\ {\isasymOBTAINS}\ A\ {\isasymBBAR}\ B{\isachardoublequote}} \\[1ex] + \isa{{\isaliteral{22}{\isachardoublequote}}disjI\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C415353554D45533E}{\isasymASSUMES}}\ A\ {\isaliteral{5C3C53484F57533E}{\isasymSHOWS}}\ A\ {\isaliteral{5C3C6F723E}{\isasymor}}\ B{\isaliteral{22}{\isachardoublequote}}} \\ + \isa{{\isaliteral{22}{\isachardoublequote}}disjI\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C415353554D45533E}{\isasymASSUMES}}\ B\ {\isaliteral{5C3C53484F57533E}{\isasymSHOWS}}\ A\ {\isaliteral{5C3C6F723E}{\isasymor}}\ B{\isaliteral{22}{\isachardoublequote}}} \\ + \isa{{\isaliteral{22}{\isachardoublequote}}disjE{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C415353554D45533E}{\isasymASSUMES}}\ A\ {\isaliteral{5C3C6F723E}{\isasymor}}\ B\ {\isaliteral{5C3C4F425441494E533E}{\isasymOBTAINS}}\ A\ {\isaliteral{5C3C424241523E}{\isasymBBAR}}\ B{\isaliteral{22}{\isachardoublequote}}} \\[1ex] - \isa{{\isachardoublequote}conjI{\isacharcolon}\ {\isasymASSUMES}\ A\ {\isasymAND}\ B\ {\isasymSHOWS}\ A\ {\isasymand}\ B{\isachardoublequote}} \\ - \isa{{\isachardoublequote}conjE{\isacharcolon}\ {\isasymASSUMES}\ A\ {\isasymand}\ B\ {\isasymOBTAINS}\ A\ {\isasymAND}\ B{\isachardoublequote}} \\[1ex] + \isa{{\isaliteral{22}{\isachardoublequote}}conjI{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C415353554D45533E}{\isasymASSUMES}}\ A\ {\isaliteral{5C3C414E443E}{\isasymAND}}\ B\ {\isaliteral{5C3C53484F57533E}{\isasymSHOWS}}\ A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B{\isaliteral{22}{\isachardoublequote}}} \\ + \isa{{\isaliteral{22}{\isachardoublequote}}conjE{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C415353554D45533E}{\isasymASSUMES}}\ A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B\ {\isaliteral{5C3C4F425441494E533E}{\isasymOBTAINS}}\ A\ {\isaliteral{5C3C414E443E}{\isasymAND}}\ B{\isaliteral{22}{\isachardoublequote}}} \\[1ex] - \isa{{\isachardoublequote}falseE{\isacharcolon}\ {\isasymASSUMES}\ {\isasymbottom}\ {\isasymSHOWS}\ A{\isachardoublequote}} \\ - \isa{{\isachardoublequote}trueI{\isacharcolon}\ {\isasymSHOWS}\ {\isasymtop}{\isachardoublequote}} \\[1ex] + \isa{{\isaliteral{22}{\isachardoublequote}}falseE{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C415353554D45533E}{\isasymASSUMES}}\ {\isaliteral{5C3C626F74746F6D3E}{\isasymbottom}}\ {\isaliteral{5C3C53484F57533E}{\isasymSHOWS}}\ A{\isaliteral{22}{\isachardoublequote}}} \\ + \isa{{\isaliteral{22}{\isachardoublequote}}trueI{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C53484F57533E}{\isasymSHOWS}}\ {\isaliteral{5C3C746F703E}{\isasymtop}}{\isaliteral{22}{\isachardoublequote}}} \\[1ex] - \isa{{\isachardoublequote}notI{\isacharcolon}\ {\isasymASSUMES}\ A\ {\isasymLongrightarrow}\ {\isasymbottom}\ {\isasymSHOWS}\ {\isasymnot}\ A{\isachardoublequote}} \\ - \isa{{\isachardoublequote}notE{\isacharcolon}\ {\isasymASSUMES}\ {\isasymnot}\ A\ {\isasymAND}\ A\ {\isasymSHOWS}\ B{\isachardoublequote}} \\[1ex] + \isa{{\isaliteral{22}{\isachardoublequote}}notI{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C415353554D45533E}{\isasymASSUMES}}\ A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C626F74746F6D3E}{\isasymbottom}}\ {\isaliteral{5C3C53484F57533E}{\isasymSHOWS}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ A{\isaliteral{22}{\isachardoublequote}}} \\ + \isa{{\isaliteral{22}{\isachardoublequote}}notE{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C415353554D45533E}{\isasymASSUMES}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ A\ {\isaliteral{5C3C414E443E}{\isasymAND}}\ A\ {\isaliteral{5C3C53484F57533E}{\isasymSHOWS}}\ B{\isaliteral{22}{\isachardoublequote}}} \\[1ex] - \isa{{\isachardoublequote}allI{\isacharcolon}\ {\isasymASSUMES}\ {\isasymAnd}x{\isachardot}\ B\ x\ {\isasymSHOWS}\ {\isasymforall}x{\isachardot}\ B\ x{\isachardoublequote}} \\ - \isa{{\isachardoublequote}allE{\isacharcolon}\ {\isasymASSUMES}\ {\isasymforall}x{\isachardot}\ B\ x\ {\isasymSHOWS}\ B\ a{\isachardoublequote}} \\[1ex] + \isa{{\isaliteral{22}{\isachardoublequote}}allI{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C415353554D45533E}{\isasymASSUMES}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ B\ x\ {\isaliteral{5C3C53484F57533E}{\isasymSHOWS}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ B\ x{\isaliteral{22}{\isachardoublequote}}} \\ + \isa{{\isaliteral{22}{\isachardoublequote}}allE{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C415353554D45533E}{\isasymASSUMES}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ B\ x\ {\isaliteral{5C3C53484F57533E}{\isasymSHOWS}}\ B\ a{\isaliteral{22}{\isachardoublequote}}} \\[1ex] - \isa{{\isachardoublequote}exI{\isacharcolon}\ {\isasymASSUMES}\ B\ a\ {\isasymSHOWS}\ {\isasymexists}x{\isachardot}\ B\ x{\isachardoublequote}} \\ - \isa{{\isachardoublequote}exE{\isacharcolon}\ {\isasymASSUMES}\ {\isasymexists}x{\isachardot}\ B\ x\ {\isasymOBTAINS}\ a\ {\isasymWHERE}\ B\ a{\isachardoublequote}} + \isa{{\isaliteral{22}{\isachardoublequote}}exI{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C415353554D45533E}{\isasymASSUMES}}\ B\ a\ {\isaliteral{5C3C53484F57533E}{\isasymSHOWS}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ B\ x{\isaliteral{22}{\isachardoublequote}}} \\ + \isa{{\isaliteral{22}{\isachardoublequote}}exE{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C415353554D45533E}{\isasymASSUMES}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ B\ x\ {\isaliteral{5C3C4F425441494E533E}{\isasymOBTAINS}}\ a\ {\isaliteral{5C3C57484552453E}{\isasymWHERE}}\ B\ a{\isaliteral{22}{\isachardoublequote}}} \end{tabular} \medskip @@ -803,7 +803,7 @@ \begin{minipage}[t]{0.4\textwidth} \isanewline \ \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}A\ {\isasymlongrightarrow}\ B{\isachardoublequoteclose}\isanewline +\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\isanewline \ \ \isacommand{proof}\isamarkupfalse% \isanewline \ \ \ \ \isacommand{assume}\isamarkupfalse% @@ -842,7 +842,7 @@ \end{minipage}\qquad\begin{minipage}[t]{0.4\textwidth} \isanewline \ \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}A\ {\isasymlongrightarrow}\ B{\isachardoublequoteclose}\ \isakeyword{and}\ A% +\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ A% \endisatagproof {\isafoldproof}% % @@ -872,7 +872,7 @@ \isatagproof \isacommand{then}\isamarkupfalse% \ \isacommand{have}\isamarkupfalse% -\ B\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ B\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% % \end{minipage}\\[3ex]\begin{minipage}[t]{0.4\textwidth} \isanewline @@ -907,7 +907,7 @@ \isatagproof \isacommand{then}\isamarkupfalse% \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}A\ {\isasymor}\ B{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C6F723E}{\isasymor}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% \isanewline \isanewline \ \ \isacommand{have}\isamarkupfalse% @@ -941,12 +941,12 @@ \isatagproof \isacommand{then}\isamarkupfalse% \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}A\ {\isasymor}\ B{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C6F723E}{\isasymor}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% % \end{minipage}\qquad\begin{minipage}[t]{0.4\textwidth} \isanewline \ \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}A\ {\isasymor}\ B{\isachardoublequoteclose}% +\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C6F723E}{\isasymor}}\ B{\isaliteral{22}{\isachardoublequoteclose}}% \endisatagproof {\isafoldproof}% % @@ -1080,12 +1080,12 @@ \isatagproof \isacommand{then}\isamarkupfalse% \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}A\ {\isasymand}\ B{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% % \end{minipage}\qquad\begin{minipage}[t]{0.4\textwidth} \isanewline \ \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}A\ {\isasymand}\ B{\isachardoublequoteclose}% +\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B{\isaliteral{22}{\isachardoublequoteclose}}% \endisatagproof {\isafoldproof}% % @@ -1115,12 +1115,12 @@ \isatagproof \isacommand{then}\isamarkupfalse% \ \isacommand{obtain}\isamarkupfalse% -\ A\ \isakeyword{and}\ B\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ A\ \isakeyword{and}\ B\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% % \end{minipage}\\[3ex]\begin{minipage}[t]{0.4\textwidth} \isanewline \ \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}{\isasymbottom}{\isachardoublequoteclose}% +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C626F74746F6D3E}{\isasymbottom}}{\isaliteral{22}{\isachardoublequoteclose}}% \endisatagproof {\isafoldproof}% % @@ -1150,24 +1150,24 @@ \isatagproof \isacommand{then}\isamarkupfalse% \ \isacommand{have}\isamarkupfalse% -\ A\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ A\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% % \end{minipage}\qquad\begin{minipage}[t]{0.4\textwidth} \isanewline \ \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}{\isasymtop}{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C746F703E}{\isasymtop}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% % \end{minipage}\\[3ex]\begin{minipage}[t]{0.4\textwidth} \isanewline \ \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}{\isasymnot}\ A{\isachardoublequoteclose}\isanewline +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ A{\isaliteral{22}{\isachardoublequoteclose}}\isanewline \ \ \isacommand{proof}\isamarkupfalse% \isanewline \ \ \ \ \isacommand{assume}\isamarkupfalse% \ A\isanewline \ \ \ \ \isacommand{then}\isamarkupfalse% \ \isacommand{show}\isamarkupfalse% -\ {\isachardoublequoteopen}{\isasymbottom}{\isachardoublequoteclose}% +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C626F74746F6D3E}{\isasymbottom}}{\isaliteral{22}{\isachardoublequoteclose}}% \endisatagproof {\isafoldproof}% % @@ -1200,7 +1200,7 @@ \end{minipage}\qquad\begin{minipage}[t]{0.4\textwidth} \isanewline \ \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}{\isasymnot}\ A{\isachardoublequoteclose}\ \isakeyword{and}\ A% +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ A{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ A% \endisatagproof {\isafoldproof}% % @@ -1230,18 +1230,18 @@ \isatagproof \isacommand{then}\isamarkupfalse% \ \isacommand{have}\isamarkupfalse% -\ B\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ B\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% % \end{minipage}\\[3ex]\begin{minipage}[t]{0.4\textwidth} \isanewline \ \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}{\isasymforall}x{\isachardot}\ B\ x{\isachardoublequoteclose}\isanewline +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ B\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline \ \ \isacommand{proof}\isamarkupfalse% \isanewline \ \ \ \ \isacommand{fix}\isamarkupfalse% \ x\isanewline \ \ \ \ \isacommand{show}\isamarkupfalse% -\ {\isachardoublequoteopen}B\ x{\isachardoublequoteclose}% +\ {\isaliteral{22}{\isachardoublequoteopen}}B\ x{\isaliteral{22}{\isachardoublequoteclose}}% \endisatagproof {\isafoldproof}% % @@ -1274,7 +1274,7 @@ \end{minipage}\qquad\begin{minipage}[t]{0.4\textwidth} \isanewline \ \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}{\isasymforall}x{\isachardot}\ B\ x{\isachardoublequoteclose}% +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ B\ x{\isaliteral{22}{\isachardoublequoteclose}}% \endisatagproof {\isafoldproof}% % @@ -1304,16 +1304,16 @@ \isatagproof \isacommand{then}\isamarkupfalse% \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}B\ a{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ {\isaliteral{22}{\isachardoublequoteopen}}B\ a{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% % \end{minipage}\\[3ex]\begin{minipage}[t]{0.4\textwidth} \isanewline \ \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}{\isasymexists}x{\isachardot}\ B\ x{\isachardoublequoteclose}\isanewline +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ B\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline \ \ \isacommand{proof}\isamarkupfalse% \isanewline \ \ \ \ \isacommand{show}\isamarkupfalse% -\ {\isachardoublequoteopen}B\ a{\isachardoublequoteclose}% +\ {\isaliteral{22}{\isachardoublequoteopen}}B\ a{\isaliteral{22}{\isachardoublequoteclose}}% \endisatagproof {\isafoldproof}% % @@ -1346,7 +1346,7 @@ \end{minipage}\qquad\begin{minipage}[t]{0.4\textwidth} \isanewline \ \ \isacommand{have}\isamarkupfalse% -\ {\isachardoublequoteopen}{\isasymexists}x{\isachardot}\ B\ x{\isachardoublequoteclose}% +\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ B\ x{\isaliteral{22}{\isachardoublequoteclose}}% \endisatagproof {\isafoldproof}% % @@ -1376,7 +1376,7 @@ \isatagproof \isacommand{then}\isamarkupfalse% \ \isacommand{obtain}\isamarkupfalse% -\ a\ \isakeyword{where}\ {\isachardoublequoteopen}B\ a{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse% +\ a\ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}B\ a{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse% % \end{minipage} %