# HG changeset patch
# User wenzelm
# Date 1307016526 -7200
# Node ID 91e229959d4ce5da5897af75001c873ea568c7eb
# Parent  ec270c6cb942249d734db9f66bd8250861ffb035
some material on "Generalized elimination and cases";

diff -r ec270c6cb942 -r 91e229959d4c doc-src/IsarRef/Thy/Synopsis.thy
--- a/doc-src/IsarRef/Thy/Synopsis.thy	Thu Jun 02 13:59:23 2011 +0200
+++ b/doc-src/IsarRef/Thy/Synopsis.thy	Thu Jun 02 14:08:46 2011 +0200
@@ -104,7 +104,7 @@
   assume "x \<noteq> y"
   note this [symmetric]
 
-  txt {* Adhoc-simplication (take care!): *}
+  txt {* Adhoc-simplification (take care!): *}
   assume "P ([] @ xs)"
   note this [simplified]
 end
@@ -954,4 +954,155 @@
   then obtain a where "a \<in> A" and "x \<in> a" ..
 end
 
+
+section {* Generalized elimination and cases *}
+
+subsection {* General elimination rules *}
+
+text {*
+  The general format of elimination rules is illustrated by the
+  following typical representatives:
+*}
+
+thm exE     -- {* local parameter *}
+thm conjE   -- {* local premises *}
+thm disjE   -- {* split into cases *}
+
+text {*
+  Combining these characteristics leads to the following general scheme
+  for elimination rules with cases:
+
+  \begin{itemize}
+
+  \item prefix of assumptions (or ``major premises'')
+
+  \item one or more cases that enable to establish the main conclusion
+    in an augmented context
+
+  \end{itemize}
+*}
+
+notepad
+begin
+  assume r:
+    "A1 \<Longrightarrow> A2 \<Longrightarrow>  (* assumptions *)
+      (\<And>x y. B1 x y \<Longrightarrow> C1 x y \<Longrightarrow> R) \<Longrightarrow>  (* case 1 *)
+      (\<And>x y. B2 x y \<Longrightarrow> C2 x y \<Longrightarrow> R) \<Longrightarrow>  (* case 2 *)
+      R  (* main conclusion *)"
+
+  have A1 and A2 sorry
+  then have R
+  proof (rule r)
+    fix x y
+    assume "B1 x y" and "C1 x y"
+    show ?thesis sorry
+  next
+    fix x y
+    assume "B2 x y" and "C2 x y"
+    show ?thesis sorry
+  qed
+end
+
+text {* Here @{text "?thesis"} is used to refer to the unchanged goal
+  statement.  *}
+
+
+subsection {* Rules with cases *}
+
+text {*
+  Applying an elimination rule to some goal, leaves that unchanged
+  but allows to augment the context in the sub-proof of each case.
+
+  Isar provides some infrastructure to support this:
+
+  \begin{itemize}
+
+  \item native language elements to state eliminations
+
+  \item symbolic case names
+
+  \item method @{method cases} to recover this structure in a
+  sub-proof
+
+  \end{itemize}
+*}
+
+print_statement exE
+print_statement conjE
+print_statement disjE
+
+lemma
+  assumes A1 and A2  -- {* assumptions *}
+  obtains
+    (case1)  x y where "B1 x y" and "C1 x y"
+  | (case2)  x y where "B2 x y" and "C2 x y"
+  sorry
+
+
+subsubsection {* Example *}
+
+lemma tertium_non_datur:
+  obtains
+    (T)  A
+  | (F)  "\<not> A"
+  by blast
+
+notepad
+begin
+  fix x y :: 'a
+  have C
+  proof (cases "x = y" rule: tertium_non_datur)
+    case T
+    from `x = y` show ?thesis sorry
+  next
+    case F
+    from `x \<noteq> y` show ?thesis sorry
+  qed
+end
+
+
+subsubsection {* Example *}
+
+text {*
+  Isabelle/HOL specification mechanisms (datatype, inductive, etc.)
+  provide suitable derived cases rules.
+*}
+
+datatype foo = Foo | Bar foo
+
+notepad
+begin
+  fix x :: foo
+  have C
+  proof (cases x)
+    case Foo
+    from `x = Foo` show ?thesis sorry
+  next
+    case (Bar a)
+    from `x = Bar a` show ?thesis sorry
+  qed
+end
+
+
+subsection {* Obtaining local contexts *}
+
+text {* A single ``case'' branch may be inlined into Isar proof text
+  via @{command obtain}.  This proves @{prop "(\<And>x. B x \<Longrightarrow> thesis) \<Longrightarrow>
+  thesis"} on the spot, and augments the context afterwards.  *}
+
+notepad
+begin
+  fix B :: "'a \<Rightarrow> bool"
+
+  obtain x where "B x" sorry
+  note `B x`
+
+  txt {* Conclusions from this context may not mention @{term x} again! *}
+  {
+    obtain x where "B x" sorry
+    from `B x` have C sorry
+  }
+  note `C`
+end
+
 end
\ No newline at end of file
diff -r ec270c6cb942 -r 91e229959d4c doc-src/IsarRef/Thy/document/Synopsis.tex
--- a/doc-src/IsarRef/Thy/document/Synopsis.tex	Thu Jun 02 13:59:23 2011 +0200
+++ b/doc-src/IsarRef/Thy/document/Synopsis.tex	Thu Jun 02 14:08:46 2011 +0200
@@ -260,7 +260,7 @@
 \ \ \isacommand{note}\isamarkupfalse%
 \ this\ {\isaliteral{5B}{\isacharbrackleft}}symmetric{\isaliteral{5D}{\isacharbrackright}}%
 \begin{isamarkuptxt}%
-Adhoc-simplication (take care!):%
+Adhoc-simplification (take care!):%
 \end{isamarkuptxt}%
 \isamarkuptrue%
 \ \ \isacommand{assume}\isamarkupfalse%
@@ -2374,6 +2374,326 @@
 %
 \endisadelimproof
 \isacommand{end}\isamarkupfalse%
+%
+\isamarkupsection{Generalized elimination and cases%
+}
+\isamarkuptrue%
+%
+\isamarkupsubsection{General elimination rules%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The general format of elimination rules is illustrated by the
+  following typical representatives:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{thm}\isamarkupfalse%
+\ exE\ \ \ \ \ %
+\isamarkupcmt{local parameter%
+}
+\isanewline
+\isacommand{thm}\isamarkupfalse%
+\ conjE\ \ \ %
+\isamarkupcmt{local premises%
+}
+\isanewline
+\isacommand{thm}\isamarkupfalse%
+\ disjE\ \ \ %
+\isamarkupcmt{split into cases%
+}
+%
+\begin{isamarkuptext}%
+Combining these characteristics leads to the following general scheme
+  for elimination rules with cases:
+
+  \begin{itemize}
+
+  \item prefix of assumptions (or ``major premises'')
+
+  \item one or more cases that enable to establish the main conclusion
+    in an augmented context
+
+  \end{itemize}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{notepad}\isamarkupfalse%
+\isanewline
+\isakeyword{begin}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{assume}\isamarkupfalse%
+\ r{\isaliteral{3A}{\isacharcolon}}\isanewline
+\ \ \ \ {\isaliteral{22}{\isachardoublequoteopen}}A{\isadigit{1}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A{\isadigit{2}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{2A}{\isacharasterisk}}\ assumptions\ {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x\ y{\isaliteral{2E}{\isachardot}}\ B{\isadigit{1}}\ x\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C{\isadigit{1}}\ x\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ R{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{2A}{\isacharasterisk}}\ case\ {\isadigit{1}}\ {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x\ y{\isaliteral{2E}{\isachardot}}\ B{\isadigit{2}}\ x\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C{\isadigit{2}}\ x\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ R{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{2A}{\isacharasterisk}}\ case\ {\isadigit{2}}\ {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \ \ \ \ R\ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{2A}{\isacharasterisk}}\ main\ conclusion\ {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isanewline
+\ \ \isacommand{have}\isamarkupfalse%
+\ A{\isadigit{1}}\ \isakeyword{and}\ A{\isadigit{2}}\ \isacommand{sorry}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{then}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ R\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}rule\ r{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \ \ \isacommand{fix}\isamarkupfalse%
+\ x\ y\isanewline
+\ \ \ \ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}B{\isadigit{1}}\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}C{\isadigit{1}}\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}thesis\ \isacommand{sorry}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{fix}\isamarkupfalse%
+\ x\ y\isanewline
+\ \ \ \ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}B{\isadigit{2}}\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}C{\isadigit{2}}\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}thesis\ \isacommand{sorry}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+\isanewline
+%
+\endisadelimproof
+\isacommand{end}\isamarkupfalse%
+%
+\begin{isamarkuptext}%
+Here \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3F}{\isacharquery}}thesis{\isaliteral{22}{\isachardoublequote}}} is used to refer to the unchanged goal
+  statement.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Rules with cases%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Applying an elimination rule to some goal, leaves that unchanged
+  but allows to augment the context in the sub-proof of each case.
+
+  Isar provides some infrastructure to support this:
+
+  \begin{itemize}
+
+  \item native language elements to state eliminations
+
+  \item symbolic case names
+
+  \item method \hyperlink{method.cases}{\mbox{\isa{cases}}} to recover this structure in a
+  sub-proof
+
+  \end{itemize}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{print{\isaliteral{5F}{\isacharunderscore}}statement}\isamarkupfalse%
+\ exE\isanewline
+\isacommand{print{\isaliteral{5F}{\isacharunderscore}}statement}\isamarkupfalse%
+\ conjE\isanewline
+\isacommand{print{\isaliteral{5F}{\isacharunderscore}}statement}\isamarkupfalse%
+\ disjE\isanewline
+\isanewline
+\isacommand{lemma}\isamarkupfalse%
+\isanewline
+\ \ \isakeyword{assumes}\ A{\isadigit{1}}\ \isakeyword{and}\ A{\isadigit{2}}\ \ %
+\isamarkupcmt{assumptions%
+}
+\isanewline
+\ \ \isakeyword{obtains}\isanewline
+\ \ \ \ {\isaliteral{28}{\isacharparenleft}}case{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ \ x\ y\ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}B{\isadigit{1}}\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}C{\isadigit{1}}\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{28}{\isacharparenleft}}case{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ \ x\ y\ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}B{\isadigit{2}}\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}C{\isadigit{2}}\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{sorry}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isamarkupsubsubsection{Example%
+}
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ tertium{\isaliteral{5F}{\isacharunderscore}}non{\isaliteral{5F}{\isacharunderscore}}datur{\isaliteral{3A}{\isacharcolon}}\isanewline
+\ \ \isakeyword{obtains}\isanewline
+\ \ \ \ {\isaliteral{28}{\isacharparenleft}}T{\isaliteral{29}{\isacharparenright}}\ \ A\isanewline
+\ \ {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{28}{\isacharparenleft}}F{\isaliteral{29}{\isacharparenright}}\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ A{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ blast%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+\isanewline
+%
+\endisadelimproof
+\isanewline
+\isacommand{notepad}\isamarkupfalse%
+\isanewline
+\isakeyword{begin}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{fix}\isamarkupfalse%
+\ x\ y\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\isanewline
+\ \ \isacommand{have}\isamarkupfalse%
+\ C\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}cases\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\ rule{\isaliteral{3A}{\isacharcolon}}\ tertium{\isaliteral{5F}{\isacharunderscore}}non{\isaliteral{5F}{\isacharunderscore}}datur{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \ \ \isacommand{case}\isamarkupfalse%
+\ T\isanewline
+\ \ \ \ \isacommand{from}\isamarkupfalse%
+\ {\isaliteral{60}{\isacharbackquoteopen}}x\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}thesis\ \isacommand{sorry}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{case}\isamarkupfalse%
+\ F\isanewline
+\ \ \ \ \isacommand{from}\isamarkupfalse%
+\ {\isaliteral{60}{\isacharbackquoteopen}}x\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ y{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}thesis\ \isacommand{sorry}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+\isanewline
+%
+\endisadelimproof
+\isacommand{end}\isamarkupfalse%
+%
+\isamarkupsubsubsection{Example%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Isabelle/HOL specification mechanisms (datatype, inductive, etc.)
+  provide suitable derived cases rules.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{datatype}\isamarkupfalse%
+\ foo\ {\isaliteral{3D}{\isacharequal}}\ Foo\ {\isaliteral{7C}{\isacharbar}}\ Bar\ foo\isanewline
+\isanewline
+\isacommand{notepad}\isamarkupfalse%
+\isanewline
+\isakeyword{begin}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{fix}\isamarkupfalse%
+\ x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ foo\isanewline
+\ \ \isacommand{have}\isamarkupfalse%
+\ C\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}cases\ x{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \ \ \isacommand{case}\isamarkupfalse%
+\ Foo\isanewline
+\ \ \ \ \isacommand{from}\isamarkupfalse%
+\ {\isaliteral{60}{\isacharbackquoteopen}}x\ {\isaliteral{3D}{\isacharequal}}\ Foo{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}thesis\ \isacommand{sorry}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{case}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}Bar\ a{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \ \ \isacommand{from}\isamarkupfalse%
+\ {\isaliteral{60}{\isacharbackquoteopen}}x\ {\isaliteral{3D}{\isacharequal}}\ Bar\ a{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}thesis\ \isacommand{sorry}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+\isanewline
+%
+\endisadelimproof
+\isacommand{end}\isamarkupfalse%
+%
+\isamarkupsubsection{Obtaining local contexts%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+A single ``case'' branch may be inlined into Isar proof text
+  via \hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}}.  This proves \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ B\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ thesis{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ thesis{\isaliteral{22}{\isachardoublequote}}} on the spot, and augments the context afterwards.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{notepad}\isamarkupfalse%
+\isanewline
+\isakeyword{begin}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{fix}\isamarkupfalse%
+\ B\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isanewline
+\ \ \isacommand{obtain}\isamarkupfalse%
+\ x\ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}B\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{sorry}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{note}\isamarkupfalse%
+\ {\isaliteral{60}{\isacharbackquoteopen}}B\ x{\isaliteral{60}{\isacharbackquoteclose}}%
+\begin{isamarkuptxt}%
+Conclusions from this context may not mention \isa{x} again!%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\ \ \isacommand{{\isaliteral{7B}{\isacharbraceleft}}}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{obtain}\isamarkupfalse%
+\ x\ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}B\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{sorry}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{from}\isamarkupfalse%
+\ {\isaliteral{60}{\isacharbackquoteopen}}B\ x{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{have}\isamarkupfalse%
+\ C\ \isacommand{sorry}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{{\isaliteral{7D}{\isacharbraceright}}}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{note}\isamarkupfalse%
+\ {\isaliteral{60}{\isacharbackquoteopen}}C{\isaliteral{60}{\isacharbackquoteclose}}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+\isanewline
+%
+\endisadelimproof
+\isacommand{end}\isamarkupfalse%
 \isanewline
 %
 \isadelimtheory