1.1 --- a/src/HOL/Real_Asymp/Manual/Real_Asymp_Doc.thy Sun Jul 22 19:55:26 2018 +0200
1.2 +++ b/src/HOL/Real_Asymp/Manual/Real_Asymp_Doc.thy Sun Jul 22 19:57:42 2018 +0200
1.3 @@ -201,7 +201,7 @@
1.4 By default, the first three terms are output and the \<open>strict\<close> option is disabled.
1.5
1.6 Note that these two commands are intended for diagnostic use only. While the central part
1.7 - of their implementation – finding a multiseries expansion and reading off the limit – are the
1.8 + of their implementation -- finding a multiseries expansion and reading off the limit -- are the
1.9 same as in the @{method real_asymp} method and therefore trustworthy, there is a small amount
1.10 of unverified code involved in pre-processing and printing (e.\,g.\ for reducing all the
1.11 different options for the \<open>limit\<close> option to the @{term at_top} case).

2.1 --- a/src/HOL/Real_Asymp/Real_Asymp_Examples.thy Sun Jul 22 19:55:26 2018 +0200
2.2 +++ b/src/HOL/Real_Asymp/Real_Asymp_Examples.thy Sun Jul 22 19:57:42 2018 +0200
2.3 @@ -100,7 +100,7 @@
2.4 unfolding f_def by real_asymp+
2.5
2.6
2.7 -subsection \<open>Asymptotic inequalities related to the Akra–Bazzi theorem\<close>
2.8 +subsection \<open>Asymptotic inequalities related to the Akra--Bazzi theorem\<close>
2.9
2.10 definition "akra_bazzi_asymptotic1 b hb e p x \<longleftrightarrow>
2.11 (1 - hb * inverse b * ln x powr -(1+e)) powr p * (1 + ln (b*x + hb*x/ln x powr (1+e)) powr (-e/2))