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Recherche/ALDLoverAB/proof/index.tex

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+We define $W^t$ as the set of processes that are winners in round $r$ at time $t$.
+
+\begin{theorem}
+    $\forall j_1, j_2 \text{ corrects}, W_{j_1} = W_{j_2}$, where $W_j$ is the set of processes that are winners in round $r$.
+\end{theorem}
+
+\begin{proof}
+    \begin{align*}
+        J = \{j_1, ..., j_n\} & \text{(set of all processes)} \\
+        B \subseteq J, B = \{b_1, ..., b_f\} & \text{(set of faulty processes)} \\
+        C \subseteq J, C = \{c_1, ..., c_{n-f}\} & \text{(set of correct processes)} \\
+        \textbf{Let's assume } \exists b_1 \in B, \exists t_0 & \text{ such that } \texttt{R-Broadcast}_{b_1}(PROP, S, r, b_1) \text{ occurs} \\
+        \Rightarrow\; & \exists K^{t_0} \subseteq C \text{ such that } \forall k \in K^{t_0}, \texttt{R-Delivered}_k(PROP, S, r, b_1) \text{ occurs} \\
+        & \wedge |K^{t_0}| = n - 2f \\
+        \Rightarrow\; & \texttt{PROVE}_k[k](<r, b_1>) \text{ is valid for all } k \in K^{t_0} \\
+        \Rightarrow\; & b_1 \not\in W^{t_0} \text{ since } \texttt{PROVE}_k[k](<r, b_1>) \text{ is valid less than } n - f \text{ times} \\
+        \text{in the same way,} & \\
+        \textbf{Let's assume } \exists L^{t_0} \subseteq C \text{ such that } & \forall l \in L^{t_0}, \texttt{R-Broadcast}_{l}(PROP, \_, r, l) \text{ occurs} \\
+        \textbf{And } \exists M^{t_0} \subseteq C \text{ such that } & \forall m \in M^{t_0}, \texttt{R-Delivered}_m(PROP, \_, r, m) \text{ occurs} \\
+        & \text{with } |L^{t_0}| = n - f \text{ and } |M^{t_0}| = n - f \\
+        \Rightarrow\; & \forall m, l : \exists (m, PROVE(<r, l>)) \in \texttt{READ}[m]() \\
+        & \textbf{And because } |M^{t_0}| \geq n - f \\
+        \Rightarrow\; & \exists O^{t_0} \subseteq M^{t_0} \text{ such that } \forall o \in O^{t_0}, W^{t_0}_o \not\ni b_1 \\
+        & \exists t_1 \geq t_0 : \forall b \in B, \texttt{PROVE}_b[b](<r, b_1>) \text{ occurs} \\
+        \Rightarrow\; & \textbf{at time } t_1, \forall k \in K : \exists (k, \texttt{PROVE}(<r, b_1>)) \in \texttt{READ}[k]() \\
+        & \textbf{And } \forall b \in B, \exists (b, \texttt{PROVE}(<r, b_1>)) \in \texttt{READ}[b]() \\
+        \Rightarrow\; & \textbf{Because } |K| + |B| = n - 2f + f = n - f \text{ the condition is satisfied} \\
+        \Rightarrow\; & W^{t_1} \ni b_1 \\
+    \end{align*}
+\end{proof}
+
+\begin{theorem}
+    % $\exist j \text{ correct } W^{t_0}$
+\end{theorem}
+
 \begin{theorem}[Integrity]
     If a message $m$ is delivered by any process, then it was previously broadcast by some process via the \texttt{AB-broadcast} primitive.
 \end{theorem}