Separation logic a tl;dr

$\{P\}\,\texttt{prog}\,\{\lambda\,v.\,Q\,v\}$

• There exists a state of type $s$
• $\texttt{prog}$ evaluates to a value $v : val$
• $P : s \to bool$
• $Q : val \to s \to bool$

$P_{1}\,\ast\,P_{2}$

$(P_{1}\,\ast\,P_{2})\,H$

• You can split state $H$ into $H_{1}$ and $H_{2}$
• $H_{1}$ and $H_{2}$ are disjoint
• $P_{1}\,H_{1}$ and $P_{2}\,H_{2}$ holds

$P_{1}\,-\!\!\ast\,P_{2}$

$(P_{1}\,-\!\!\ast\,P_{2})\,H$

• You can split state $H$ into $H_{1}$ and $H_{2}$
• $H_{1}$ and $H_{2}$ are disjoint
• if $P_{1}\,H_{1}$ holds then $P_{2}\,(H_{1}\,\cup\,H_{2})$ holds

$ \{a \mapsto 1 \ast b \mapsto 2 \ast c \mapsto 3\}$

$ a := b + c $

$ \{\lambda\_.\,a \mapsto 5 \ast b \mapsto 2 \ast c \mapsto 3\} $

$ \{\top\}$

$ a := b + c $

$ \{\lambda\_.\,(b \mapsto 2 \ast c \mapsto 3) -\!\!\ast\, a \mapsto 5 \} $