--- a/src/Doc/Tutorial/Misc/natsum.thy Wed Dec 26 16:07:28 2018 +0100
+++ b/src/Doc/Tutorial/Misc/natsum.thy Wed Dec 26 16:25:20 2018 +0100
@@ -2,7 +2,7 @@
theory natsum imports Main begin
(*>*)
text\<open>\noindent
-In particular, there are @{text"case"}-expressions, for example
+In particular, there are \<open>case\<close>-expressions, for example
@{term[display]"case n of 0 => 0 | Suc m => m"}
primitive recursion, for example
\<close>
@@ -38,17 +38,17 @@
\cdx{max}, \isadxboldpos{\isasymle}{$HOL2arithrel} and
\isadxboldpos{<}{$HOL2arithrel} are overloaded: they are available
not just for natural numbers but for other types as well.
- For example, given the goal @{text"x + 0 = x"}, there is nothing to indicate
+ For example, given the goal \<open>x + 0 = x\<close>, there is nothing to indicate
that you are talking about natural numbers. Hence Isabelle can only infer
- that @{term x} is of some arbitrary type where @{text 0} and @{text"+"} are
+ that @{term x} is of some arbitrary type where \<open>0\<close> and \<open>+\<close> are
declared. As a consequence, you will be unable to prove the
goal. To alert you to such pitfalls, Isabelle flags numerals without a
fixed type in its output: @{prop"x+0 = x"}. (In the absence of a numeral,
it may take you some time to realize what has happened if \pgmenu{Show
Types} is not set). In this particular example, you need to include
- an explicit type constraint, for example @{text"x+0 = (x::nat)"}. If there
+ an explicit type constraint, for example \<open>x+0 = (x::nat)\<close>. If there
is enough contextual information this may not be necessary: @{prop"Suc x =
- x"} automatically implies @{text"x::nat"} because @{term Suc} is not
+ x"} automatically implies \<open>x::nat\<close> because @{term Suc} is not
overloaded.
For details on overloading see \S\ref{sec:overloading}.
@@ -57,20 +57,20 @@
\end{warn}
\begin{warn}
The symbols \isadxboldpos{>}{$HOL2arithrel} and
- \isadxboldpos{\isasymge}{$HOL2arithrel} are merely syntax: @{text"x > y"}
- stands for @{prop"y < x"} and similary for @{text"\<ge>"} and
- @{text"\<le>"}.
+ \isadxboldpos{\isasymge}{$HOL2arithrel} are merely syntax: \<open>x > y\<close>
+ stands for @{prop"y < x"} and similary for \<open>\<ge>\<close> and
+ \<open>\<le>\<close>.
\end{warn}
\begin{warn}
- Constant @{text"1::nat"} is defined to equal @{term"Suc 0"}. This definition
+ Constant \<open>1::nat\<close> is defined to equal @{term"Suc 0"}. This definition
(see \S\ref{sec:ConstDefinitions}) is unfolded automatically by some
- tactics (like @{text auto}, @{text simp} and @{text arith}) but not by
+ tactics (like \<open>auto\<close>, \<open>simp\<close> and \<open>arith\<close>) but not by
others (especially the single step tactics in Chapter~\ref{chap:rules}).
If you need the full set of numerals, see~\S\ref{sec:numerals}.
\emph{Novices are advised to stick to @{term"0::nat"} and @{term Suc}.}
\end{warn}
-Both @{text auto} and @{text simp}
+Both \<open>auto\<close> and \<open>simp\<close>
(a method introduced below, \S\ref{sec:Simplification}) prove
simple arithmetic goals automatically:
\<close>
@@ -81,7 +81,7 @@
text\<open>\noindent
For efficiency's sake, this built-in prover ignores quantified formulae,
many logical connectives, and all arithmetic operations apart from addition.
-In consequence, @{text auto} and @{text simp} cannot prove this slightly more complex goal:
+In consequence, \<open>auto\<close> and \<open>simp\<close> cannot prove this slightly more complex goal:
\<close>
lemma "m \<noteq> (n::nat) \<Longrightarrow> m < n \<or> n < m"
@@ -89,10 +89,10 @@
text\<open>\noindent The method \methdx{arith} is more general. It attempts to
prove the first subgoal provided it is a \textbf{linear arithmetic} formula.
-Such formulas may involve the usual logical connectives (@{text"\<not>"},
-@{text"\<and>"}, @{text"\<or>"}, @{text"\<longrightarrow>"}, @{text"="},
-@{text"\<forall>"}, @{text"\<exists>"}), the relations @{text"="},
-@{text"\<le>"} and @{text"<"}, and the operations @{text"+"}, @{text"-"},
+Such formulas may involve the usual logical connectives (\<open>\<not>\<close>,
+\<open>\<and>\<close>, \<open>\<or>\<close>, \<open>\<longrightarrow>\<close>, \<open>=\<close>,
+\<open>\<forall>\<close>, \<open>\<exists>\<close>), the relations \<open>=\<close>,
+\<open>\<le>\<close> and \<open><\<close>, and the operations \<open>+\<close>, \<open>-\<close>,
@{term min} and @{term max}. For example,\<close>
lemma "min i (max j (k*k)) = max (min (k*k) i) (min i (j::nat))"
@@ -107,19 +107,19 @@
(*<*)oops(*>*)
text\<open>\noindent
-is not proved by @{text arith} because the proof relies
+is not proved by \<open>arith\<close> because the proof relies
on properties of multiplication. Only multiplication by numerals (which is
the same as iterated addition) is taken into account.
-\begin{warn} The running time of @{text arith} is exponential in the number
+\begin{warn} The running time of \<open>arith\<close> is exponential in the number
of occurrences of \ttindexboldpos{-}{$HOL2arithfun}, \cdx{min} and
\cdx{max} because they are first eliminated by case distinctions.
-If @{text k} is a numeral, \sdx{div}~@{text k}, \sdx{mod}~@{text k} and
-@{text k}~\sdx{dvd} are also supported, where the former two are eliminated
+If \<open>k\<close> is a numeral, \sdx{div}~\<open>k\<close>, \sdx{mod}~\<open>k\<close> and
+\<open>k\<close>~\sdx{dvd} are also supported, where the former two are eliminated
by case distinctions, again blowing up the running time.
-If the formula involves quantifiers, @{text arith} may take
+If the formula involves quantifiers, \<open>arith\<close> may take
super-exponential time and space.
\end{warn}
\<close>