[FMFP] Mostly finish exercises section

semester4/fmfp/parts/06_exercises/02_fm/00_imp.tex

@@ -1 +1,9 @@
-\subsubsection{IMP}
+\subsubsection{IMP Proofs}
+\begin{examdetails}
+    Typically, the exam contains some operational semantics tasks,
+    rarely proofs of properties of IMP.
+\end{examdetails}
+The property proofs can be acheived either using a direct proof of the implication, or using an induction proof.
+The latter of which is more common and we typically define $P$ in such a way that we bind everything apart from variables, natural numbers and constant values
+using quantifiers, then prove $\forall n \in \N . P(n)$, if we have a free natural numbers.
+Before the definition of $P$ we typically state ``Let $<$variables, constants$>$ be arbitrary'', as with any induction proof.

semester4/fmfp/parts/06_exercises/02_fm/01_operational-semantics.tex

@@ -1,5 +1,34 @@
 \subsubsection{Operational Semantics}
-\paragraph{Big Step Semantics}
+\paragraph{Big Step Semantics (Natural Semantics, NS)}
+Proving something using the big step semantics rules tends to be fairly easy if you understand how natural deduction and the like work.
+The concept is very similar, just with other rules. It is advisable to name each sub-statement in the main statement, to reduce the required amount writing.
+
+Sometimes, Induction on (the shape of) Proof Trees may be required, for that, see Section~\ref{sec:induction-on-proof-trees}
+
+Where it gets more interesting and challenging, is providing rules in the big step semantics.
+Here, it is important to think about the different cases that could be introduced by the statement.
+
+For instance, for loops, there will (almost certainly) be two cases, one in which some boolean expression (denoted $\cB \llbracket e \rrbracket \sigma$) is \texttt{ff},
+the other in which it is \texttt{tt}.
+For these, you need to make sure to state the side condition (i.e. that the boolean expression is true or false, respectively).
+
+In addition, think thoroughly about how to define the states of the rules. For instance, for a \texttt{time} statement, that counts the number of assignments,
+the state we map to would ideally be defined as $\sigma' [x \mapsto n]$, where $n$ is the number of assignments.
 
 
-\paragraph{Small Step Semantics}
+\paragraph{Small Step Semantics (Structural Operational Semantics, SOS)}
+Small Step semantics proofs tend to be a bit more challenging. Unlike Natural Semantics, here we can't create a single derivation tree, but one per step.
+This means, that we do one single step transition at a time, denoted $\rightarrow_1^1$, and justify each step with a proof tree and SO Semantics (SOS).
+Typically, only \textit{some} of the statements are asked to be proven in the exam, as proving all is time intensive due to the typically fairly
+large number of steps in the proofs.
+For the proof trees, they need to be complete for each transition that we prove, i.e. no hypothesis in any branches.
+
+Defining rules here is similar in concept as with the Natural Semantics.
+We want to make sure, to also consider the effects it has on other rules, as with Natural Semantics.
+
+The sole difference here is that we return \textit{both} the next statement to be evaluated \textit{and} the next state,
+unlike with Natural Semantics, where we only return the next state.
+
+
+\paragraph{Other proofs}
+Induction proofs for these two are also outlined in Section~\ref{sec:induction-on-proof-trees} and subsequent subsections.

semester4/fmfp/parts/06_exercises/02_fm/02_axiomatic-semantics.tex

@@ -1 +1,11 @@
 \subsubsection{Axiomatic Semantics}
+Here, we need to find pre- and postconditions for expressions and prove that they are correct.
+
+To do the proofs, we again apply transition rules, as was the case already with operational semantics.
+Contrary to those however, we have pre- and postconditions, which we typically need to define ourselves.
+
+This typically involves finding a loop invariant that holds before and after each iteration of the loop.
+This invariant should mention every variable used in the loop.
+Any other variable should also be mentioned in it. The loop \textit{variant} may also be added for proving termination.
+A typical for-loop loop variant would be \texttt{n - x}, as the next value of the loop variant has to be lower than the previous one.
+Of course, if \texttt{x} is decreasing, it itself can become the variant, as it fulfils the condition.

semester4/fmfp/parts/06_exercises/02_fm/03_modelling.tex

@@ -1 +1,12 @@
 \subsubsection{Modelling}
+Many of the tasks here are pretty straight forward, converting IMP into Promela,
+and putting an \texttt{assert} (or more) into the \texttt{init} block, to check if the required state is reached.
+
+Note that for non-determinism, we use Promela conditionals without conditions.
+
+For parallelism, we can use the \texttt{run} keyword for a \texttt{proctype}, then we wait for the processes to terminate using
+\texttt{\_nr\_pr == 1} and do our assertion.
+
+
+If we are given a text description only however, things get more challenging, as we need to model the properties given there.
+A possible simplification of the process may be to model things one property at a time, then combine them as needed.

semester4/fmfp/parts/06_exercises/02_fm/04_ltl.tex

@@ -1 +1,2 @@
 \subsubsection{Linear Time Properties (LTL)}
+We refer to Section~\ref{sec:ltl} for intuition, as these tend to mostly be intuition exercises.