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\begin{isabellebody}%
\def\isabellecontext{Event}%
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\isadelimtheory
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\endisadelimtheory
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\isatagtheory
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\isadelimproof
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\isamarkupsection{Event Traces \label{sec:events}%
}
\isamarkuptrue%
%
\begin{isamarkuptext}%
The system's behaviour is formalized as a set of traces of
\emph{events}.  The most important event, \isa{Says\ A\ B\ X}, expresses
$A\to B : X$, which is the attempt by~$A$ to send~$B$ the message~$X$.
A trace is simply a list, constructed in reverse
using~\isa{{\isaliteral{23}{\isacharhash}}}.  Other event types include reception of messages (when
we want to make it explicit) and an agent's storing a fact.

Sometimes the protocol requires an agent to generate a new nonce. The
probability that a 20-byte random number has appeared before is effectively
zero.  To formalize this important property, the set \isa{used\ evs}
denotes the set of all items mentioned in the trace~\isa{evs}.
The function \isa{used} has a straightforward
recursive definition.  Here is the case for \isa{Says} event:
\begin{isabelle}%
\ \ \ \ \ used\ {\isaliteral{28}{\isacharparenleft}}Says\ A\ B\ X\ {\isaliteral{23}{\isacharhash}}\ evs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ parts\ {\isaliteral{7B}{\isacharbraceleft}}X{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ used\ evs%
\end{isabelle}

The function \isa{knows} formalizes an agent's knowledge.  Mostly we only
care about the spy's knowledge, and \isa{knows\ Spy\ evs} is the set of items
available to the spy in the trace~\isa{evs}.  Already in the empty trace,
the spy starts with some secrets at his disposal, such as the private keys
of compromised users.  After each \isa{Says} event, the spy learns the
message that was sent:
\begin{isabelle}%
\ \ \ \ \ knows\ Spy\ {\isaliteral{28}{\isacharparenleft}}Says\ A\ B\ X\ {\isaliteral{23}{\isacharhash}}\ evs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ insert\ X\ {\isaliteral{28}{\isacharparenleft}}knows\ Spy\ evs{\isaliteral{29}{\isacharparenright}}%
\end{isabelle}
Combinations of functions express other important
sets of messages derived from~\isa{evs}:
\begin{itemize}
\item \isa{analz\ {\isaliteral{28}{\isacharparenleft}}knows\ Spy\ evs{\isaliteral{29}{\isacharparenright}}} is everything that the spy could
learn by decryption
\item \isa{synth\ {\isaliteral{28}{\isacharparenleft}}analz\ {\isaliteral{28}{\isacharparenleft}}knows\ Spy\ evs{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}} is everything that the spy
could generate
\end{itemize}%
\end{isamarkuptext}%
\isamarkuptrue%
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\end{isabellebody}%
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