# HG changeset patch # User wenzelm # Date 1711021671 -3600 # Node ID 82aaa0d8fc3bf34bdd1353f09231e7f49ecd66e7 # Parent 67d28b35c5d8e25cf6d7ab2929b9d0fdc6dc1941 isabelle update -u cite; diff -r 67d28b35c5d8 -r 82aaa0d8fc3b src/HOL/Analysis/Set_Integral.thy --- a/src/HOL/Analysis/Set_Integral.thy Wed Mar 20 21:12:49 2024 +0100 +++ b/src/HOL/Analysis/Set_Integral.thy Thu Mar 21 12:47:51 2024 +0100 @@ -1745,11 +1745,11 @@ subsection \Averaging Theorem\ text \We aim to lift results from the real case to arbitrary Banach spaces. - Our fundamental tool in this regard will be the averaging theorem. The proof of this theorem is due to Serge Lang (Real and Functional Analysis) \cite{Lang_1993}. + Our fundamental tool in this regard will be the averaging theorem. The proof of this theorem is due to Serge Lang (Real and Functional Analysis) \<^cite>\"Lang_1993"\. The theorem allows us to make statements about a function's value almost everywhere, depending on the value its integral takes on various sets of the measure space.\ text \Before we introduce and prove the averaging theorem, we will first show the following lemma which is crucial for our proof. - While not stated exactly in this manner, our proof makes use of the characterization of second-countable topological spaces given in the book General Topology by Ryszard Engelking (Theorem 4.1.15) \cite{engelking_1989}. + While not stated exactly in this manner, our proof makes use of the characterization of second-countable topological spaces given in the book General Topology by Ryszard Engelking (Theorem 4.1.15) \<^cite>\"engelking_1989"\. (Engelking's book \emph{General Topology})\ lemma balls_countable_basis: diff -r 67d28b35c5d8 -r 82aaa0d8fc3b src/HOL/Library/Centered_Division.thy --- a/src/HOL/Library/Centered_Division.thy Wed Mar 20 21:12:49 2024 +0100 +++ b/src/HOL/Library/Centered_Division.thy Thu Mar 21 12:47:51 2024 +0100 @@ -15,7 +15,7 @@ \noindent The following specification of division on integers centers the modulus around zero. This is useful e.g.~to define division on Gauss numbers. - N.b.: This is not mentioned \cite{leijen01}. + N.b.: This is not mentioned \<^cite>\"leijen01"\. \ definition centered_divide :: \int \ int \ int\ (infixl \cdiv\ 70) diff -r 67d28b35c5d8 -r 82aaa0d8fc3b src/HOL/Library/Signed_Division.thy --- a/src/HOL/Library/Signed_Division.thy Wed Mar 20 21:12:49 2024 +0100 +++ b/src/HOL/Library/Signed_Division.thy Thu Mar 21 12:47:51 2024 +0100 @@ -48,7 +48,7 @@ end text \ - \noindent The following specification of division is named ``T-division'' in \cite{leijen01}. + \noindent The following specification of division is named ``T-division'' in \<^cite>\"leijen01"\. It is motivated by ISO C99, which in turn adopted the typical behavior of hardware modern in the beginning of the 1990ies; but note ISO C99 describes the instance on machine words, not mathematical integers. diff -r 67d28b35c5d8 -r 82aaa0d8fc3b src/HOL/Library/Word.thy --- a/src/HOL/Library/Word.thy Wed Mar 20 21:12:49 2024 +0100 +++ b/src/HOL/Library/Word.thy Thu Mar 21 12:47:51 2024 +0100 @@ -626,7 +626,7 @@ text \ The following specification of word division just lifts the pre-existing - division on integers named ``F-Division'' in \cite{leijen01}. + division on integers named ``F-Division'' in \<^cite>\"leijen01"\. \ instantiation word :: (len) semiring_modulo