info/exercises/src/ex-04/ex/semantics.tex

+
+
+\begin{exercise}{}
+
+  Consider the following expression language over naturals, and a \emph{halving}
+  operator:
+  %
+  \begin{align*}
+    expr ::= \half(expr) \mid expr + expr \mid \num
+  \end{align*}
+  where \(\num\) is any natural number constant \(\ge 0\).
+
+  We will design the operational semantics of this language. The semantics
+  should define rules that apply to as many expressions as possible, while being
+  subjected to the following safety conditions:
+  \begin{itemize}
+    \item the semantics should \emph{not} permit halving unless the argument is even
+    \item they should evaluate operands from left-to-right
+  \end{itemize}
+
+  Of the given rules below, choose a \emph{minimal} set that satisfies the
+  conditions above. A set is \emph{not} minimal if removing any rule does not
+  change the set of expressions that can be evaluated by the semantics, i.e. the
+  domain of \(\leadsto, \{x \mid \exists y.\; x \leadsto y\}\), remains
+  unchanged. The removed rule is said to be \emph{redundant}.
+
+  
+  % \begin{multicols}{2}
+    \allowdisplaybreaks
+    \addtolength{\jot}{2ex}
+    \begin{gather*}
+      \AxiomC{\(e \leadsto e'\)}
+      \UnaryInfC{\(\half(e) \leadsto e'\)}
+      \DisplayProof \tag{A} \\
+      %
+      \AxiomC{\(n\) is a value}
+      \AxiomC{\(n = 2k\)}
+      \BinaryInfC{\(\half(n) \leadsto k\)}
+      \DisplayProof \tag{B} \\
+      %
+      \AxiomC{\(n\) is a value}
+      \UnaryInfC{\(\half(n) \leadsto \lfloor \frac{n}{2} \rfloor\)}
+      \DisplayProof \tag{C} \\
+      %
+      \AxiomC{\(\half(e) \leadsto \half(e')\)}
+      \UnaryInfC{\(\half(e) \leadsto e'\)}
+      \DisplayProof \tag{D} \\
+      %
+      \AxiomC{\(e \leadsto e'\)}
+      \UnaryInfC{\(\half(e) \leadsto \half(e')\)}
+      \DisplayProof \tag{E} \\
+      %
+      \AxiomC{\(e' \leadsto \half(e)\)}
+      \UnaryInfC{\(half(e) \leadsto e'\)}
+      \DisplayProof \tag{F} \\
+      %
+      \AxiomC{\(n_1\) is a value}
+      \AxiomC{\(n_2\) is a value}
+      \AxiomC{\(n_1 + n_2 = k\)}
+      \AxiomC{\(n_1\) is odd}
+      \QuaternaryInfC{\(n_1 + n_2 \leadsto k\)}
+      \DisplayProof \tag{G} \\
+      %
+      \AxiomC{\(e \leadsto e'\)}
+      \AxiomC{\(n\) is a value}
+      \BinaryInfC{\(n + e \leadsto n + e'\)}
+      \DisplayProof \tag{H} \\
+      %
+      \AxiomC{\(e_2 \leadsto e_2'\)}
+      \UnaryInfC{\(e_1 + e_2 \leadsto e_1 + e_2'\)}
+      \DisplayProof \tag{I} \\
+      %
+      \AxiomC{\(n_1\) is a value}
+      \AxiomC{\(n_2\) is a value}
+      \AxiomC{\(n_1 + n_2 = k\)}
+      \AxiomC{\(n_1, n_2\) are even}
+      \QuaternaryInfC{\(n_1 + n_2 \leadsto k\)}
+      \DisplayProof \tag{J} \\
+      %
+      \AxiomC{\(n_1\) is a value}
+      \AxiomC{\(n_2\) is a value}
+      \AxiomC{\(n_1 + n_2 = k\)}
+      \TrinaryInfC{\(n_1 + n_2 \leadsto k\)}
+      \DisplayProof \tag{K} \\
+      %
+      \AxiomC{\(e_1 \leadsto e_1'\)}
+      \UnaryInfC{\(e_1 + e_2 \leadsto e_1' + e_2\)}
+      \DisplayProof \tag{L} 
+    \end{gather*}
+  % \end{multicols}
+
+  \begin{solution}
+    A possible such minimal set of rules is \(\{B, E, H, K, L\}\). % TODO:
+
+    On what happens when the other rules are added to this set:
+    \begin{itemize}
+      \item A: incorrect; allows deducing \(\half(\half(10)) \leadsto 5\) with
+      rule B.
+      \item C: incorrect; allows deducing \(\half(3) \leadsto 1\).
+      \item D: incorrect; allows deducing \(\half(\half(10)) \leadsto 5\) with
+      rules B and E.
+      \item F: redundant; reverses a reduction.
+      \item G: redundant; special case of rule K.
+      \item I: incorrect; does not reduce the expression left-to-right.
+      \item J: redundant; special case of rule K.
+    \end{itemize}
+  \end{solution}
+
+\end{exercise}
+
+\begin{exercise}{}
+
+  Consider a simple programming language with integer arithmetic, boolean
+  expressions, and user-defined functions:
+
+  \begin{align*}
+    expr ::= &~ true \mid false \mid \num \\
+             &~ expr == expr \mid expr + expr \\
+             &~ expr ~\&\& ~expr \mid if ~(expr) ~expr ~else ~expr \\
+             &~ f(expr, \ldots, expr) \mid x
+  \end{align*}
+  %
+  where \(f\) represents a (user-defined) function, \(x\) represents a
+  variable, and \(\num\) represents an integer.
+
+  \begin{enumerate}
+    \item Inductively define a substitution operation for the terms in this
+      language, which replaces every free occurrence of a variable \(x\) with a
+      given expression \(e\).
+
+      The rule for substitution in an addition is provided as an example. Here,
+      \(t[x := e]\) represents the term \(t\), with every free occurrence of \(x\)
+      simultaneously replaced by \(e\).
+
+      \begin{equation*}
+        \AxiomC{\(t_1[x := e] \to t_1'\)}
+        \AxiomC{\(t_2[x := e] \to t_2'\)}
+        \BinaryInfC{\(t_1 + t_2[x := e] \to t_1' + t_2'\)}
+        \DisplayProof
+      \end{equation*}
+
+    \item Write the rules for the operational semantics for this language,
+    assuming \emph{call-by-name} semantics for function calls. In call-by-name
+    semantics, function arguments are not evaluated before the call. Instead,
+    the parameters are merely substituted into the function body. You may assume
+    that function parameters are named distinctly from variables in the program.
+
+    \item Under the following environment (with function names, parameters, and
+    bodies):
+      \begin{gather*}
+        (sum, [x], if ~ (x == 0) ~ then ~ 0 ~ else ~ x + sum(x + (-1))) \\
+        (rec, [~], rec()) \\
+        (default, [b, x], if ~ b ~ then ~ x ~ else ~ 0)
+      \end{gather*}
+
+      evaluate each of the following expressions, showing the derivations:
+      \begin{enumerate}
+        \item \(sum(2)\)
+        \item \(if ~(1 == 2) ~ then ~ 3 ~ else ~ 4\)
+        \item \(sum(sum(0))\)
+        \item \(rec()\)
+        \item \(default(false, rec())\)
+      \end{enumerate}
+
+      How would the evaluations in each case change if we used \emph{call-by-value}
+      semantics instead?
+  \end{enumerate}
+
+  \begin{solution}
+    \allowdisplaybreaks
+
+    \begin{enumerate}
+      \item Substitution rules:
+      \begin{gather*}
+        \AxiomC{}
+        \UnaryInfC{\(true[x := e] \to true\)}
+        \DisplayProof \\
+        %
+        \AxiomC{}
+        \UnaryInfC{\(false[x := e] \to false\)}
+        \DisplayProof \\
+        %
+        \AxiomC{}
+        \UnaryInfC{\(\num[x := e] \to \num\)}
+        \DisplayProof \\
+        %
+        \AxiomC{\(t_1[x := e] \to t_1'\)}
+        \AxiomC{\(t_2[x := e] \to t_2'\)}
+        \BinaryInfC{\(t_1 == t_2[x := e] \to t_1' == t_2'\)}
+        \DisplayProof \\
+        %
+        \AxiomC{\(t_1[x := e] \to t_1'\)}
+        \AxiomC{\(t_2[x := e] \to t_2'\)}
+        \BinaryInfC{\(t_1 + t_2[x := e] \to t_1' + t_2'\)}
+        \DisplayProof \\
+        %
+        \AxiomC{\(t_1[x := e] \to t_1'\)}
+        \AxiomC{\(t_2[x := e] \to t_2'\)}
+        \BinaryInfC{\(t_1 ~\&\& ~t_2[x := e] \to t_1' ~\&\& ~t_2'\)}
+        \DisplayProof \\
+        %
+        \AxiomC{\(t_1[x := e] \to t_1'\)}
+        \AxiomC{\(t_2[x := e] \to t_2'\)}
+        \AxiomC{\(t_3[x := e] \to t_3'\)}
+        \TrinaryInfC{\(if ~(t_1) ~t_2 ~else ~t_3[x := e] \to if ~(t_1') ~t_2' ~else ~t_3'\)}
+        \DisplayProof \\
+        %
+        \AxiomC{\(t_1[x := e] \to t_1'\)}
+        \AxiomC{\(\ldots\)}
+        \AxiomC{\(t_n[x := e] \to t_3'\)}
+        \TrinaryInfC{\(f(t_1, \ldots, t_n)[x := e] \to f(t_1', \ldots, t_n')\)}
+        \DisplayProof \\
+        %
+        \AxiomC{}
+        \UnaryInfC{\(x[x := e] \to e\)}
+        \DisplayProof \\
+        %
+        \AxiomC{\(x \neq y\)}
+        \UnaryInfC{\(y[x := e] \to y\)}
+        \DisplayProof
+      \end{gather*}
+
+      \item Operational semantics:
+      \begin{itemize}
+        \item Equality:
+          \begin{gather*}
+            \AxiomC{\(n_1, n_2\) are integer values}
+            \AxiomC{\(n_1 = n_2\)}
+            \BinaryInfC{\(n_1 == n_2 \leadsto true\)}
+            \DisplayProof \\
+            %
+            \AxiomC{\(n_1, n_2\) are integer values}
+            \AxiomC{\(n_1 \neq n_2\)}
+            \BinaryInfC{\(n_1 == n_2 \leadsto false\)}
+            \DisplayProof
+          \end{gather*}
+        \item Addition:
+          \begin{gather*}
+            \AxiomC{\(t_1 \leadsto t_1'\)}
+            \UnaryInfC{\(t_1 + t_2 \leadsto t_1' + t_2\)}
+            \DisplayProof \\
+            %
+            \AxiomC{\(n\) is an integer value}
+            \AxiomC{\(t_2 \leadsto t_2'\)}
+            \BinaryInfC{\(n + t_2 \leadsto n + t_2'\)}
+            \DisplayProof \\
+            %
+            \AxiomC{\(n_1, n_2\) are integer values}
+            \AxiomC{\(n_1 + n_2 = k\)}
+            \BinaryInfC{\(n_1 + n_2 \leadsto k\)}
+            \DisplayProof
+          \end{gather*}
+        \item Conjunction:
+          \begin{gather*}
+            \AxiomC{\(t_1 \leadsto t_1'\)}
+            \UnaryInfC{\(t_1 ~\&\& ~t_2 \leadsto t_1' ~\&\& ~t_2\)}
+            \DisplayProof \\
+            %
+            \AxiomC{}
+            \UnaryInfC{\(true ~\&\&~ t \leadsto t\)}
+            \DisplayProof \\
+            %
+            \AxiomC{}
+            \UnaryInfC{\(false ~\&\&~ t \leadsto false\)}
+            \DisplayProof
+          \end{gather*}
+        \item Conditionals:
+          \begin{gather*}
+            \AxiomC{\(t_1 \leadsto t_1'\)}
+            \UnaryInfC{\(if ~(t_1) ~t_2 ~else ~t_3 \leadsto if ~(t_1') ~t_2 ~else ~t_3\)}
+            \DisplayProof \\
+            %
+            \AxiomC{}
+            \UnaryInfC{\(if ~(true) ~t_2 ~else ~t_3 \leadsto t_2\)}
+            \DisplayProof \\
+            %
+            \AxiomC{}
+            \UnaryInfC{\(if ~(false) ~t_2 ~else ~t_3 \leadsto t_3\)}
+            \DisplayProof
+          \end{gather*}
+        \item Function call:
+          \begin{gather*}
+            \alwaysNoLine
+            \AxiomC{\(b_0\) is the body of \(f\)}
+            \UnaryInfC{\((x_1, \ldots, x_n)\) are parameters of \(f\)}
+            \UnaryInfC{\(b_0[x_1 := t_1] \to b_1 \quad \ldots \quad b_{n-1}[x_n := t_n] \to b_n\)}
+            \alwaysSingleLine
+            \UnaryInfC{\(f(t_1, \ldots, t_n) \leadsto b_n\)}
+            \DisplayProof
+          \end{gather*}
+      \end{itemize}
+
+      \item Evaluations:
+      \begin{enumerate}
+        \item \(sum(2) \leadsto 3\):
+          \begin{align*}
+            &sum(2)\\
+            &\leadsto if ~(2 == 0) ~ then ~ 0 ~ else ~ 2 + sum(2 + (-1))\\
+            &\leadsto if ~(false) ~ then ~ 0 ~ else ~ 2 + sum(2 + (-1))\\
+            &\leadsto 2 + sum(2 + (-1))\\
+            &\leadsto 2 + if ~((2 + (-1)) == 0) ~ then ~ 0 ~ else ~ 1 + sum(2 + (-1))\\
+            &\leadsto 2 + if ~(1 == 0) ~ then ~ 0 ~ else ~ 1 + sum(2 + (-1))\\
+            &\leadsto 2 + if ~(false) ~ then ~ 0 ~ else ~ 1 + sum(2 + (-1) + (-1))\\
+            &\leadsto 2 + (1 + sum(2 + (-1) + (-1)))\\
+            & \ldots \\
+            &\leadsto 2 + (1 + 0) \\
+            &\leadsto 2 + 1 \\
+            &\leadsto 3
+          \end{align*}
+        \item \(sum(sum(0)) \leadsto 0\):
+        \begin{align*}
+          &sum(sum(0))\\
+          &\leadsto if ~(sum(0) == 0) ~ then ~ 0 ~ else ~ 0 + sum(sum(0) + (-1))\\
+          &\leadsto \ldots \textit{(expand \(sum(0)\) in the conditional)} \\
+          &\leadsto if ~(0 == 0) ~ then ~ 0 ~ else ~ 0 + sum(sum(0) + (-1))\\
+          &\leadsto if ~(true) ~ then ~ 0 ~ else ~ 0 + sum(sum(0) + (-1))\\
+          &\leadsto 0
+        \end{align*}
+        \item \(if ~(1 == 2) ~ then ~ 3 ~ else ~ 4 \leadsto 4\).
+        \item \(rec() \leadsto rec()\) (infinite loop).
+        \item \(default(false, rec()) \leadsto 0\).
+      \end{enumerate}
+
+      Under call-by-value-semantics, the structure of the evaluations would be
+      different. In \(sum(sum(0))\), we would evaluate the inner \(sum(0)\) to
+      \(0\) before evaluating the outer \(sum(\cdot)\). In \(default(false,
+      rec())\), we would need to evaluate \(rec()\), which would lead to an
+      infinite loop.
+    \end{enumerate}
+        
+  \end{solution}
+  
+\end{exercise}

info/exercises/src/ex-04/ex/types.tex

+
+% perform actual type derivation of some terms
+
+\begin{exercise}{}
+
+  Consider the following type system for a language with integers, conditionals, pairs, and
+  functions:
+
+  \allowdisplaybreaks
+  \addtolength{\jot}{0.5em}
+  \begin{gather*}
+    \AxiomC{\(n\) is an integer literal}
+    \UnaryInfC{\(\Gamma \vdash n : \lstinline|Int|\)}
+    \DisplayProof \quad
+    % variable
+    \AxiomC{\(x : \tau \in \Gamma\)}
+    \UnaryInfC{\(\Gamma \vdash x : \tau\)}
+    \DisplayProof \\
+    %
+    % addition
+    \AxiomC{\(\Gamma \vdash e_1 : \lstinline|Int|\)}
+    \AxiomC{\(\Gamma \vdash e_2 : \lstinline|Int|\)}
+    \BinaryInfC{\(\Gamma \vdash e_1 + e_2 : \lstinline|Int|\)}
+    \DisplayProof \\
+    %
+    % multiplication
+    \AxiomC{\(\Gamma \vdash e_1 : \lstinline|Int|\)}
+    \AxiomC{\(\Gamma \vdash e_2 : \lstinline|Int|\)}
+    \BinaryInfC{\(\Gamma \vdash e_1 \times e_2 : \lstinline|Int|\)}
+    \DisplayProof \\
+    %
+    % booleans
+    \AxiomC{\(b\) is a boolean literal}
+    \UnaryInfC{\(\Gamma \vdash b : \lstinline|Bool|\)}
+    \DisplayProof \quad
+    \AxiomC{\(\Gamma \vdash e : \lstinline|Bool|\)}
+    \UnaryInfC{\(\Gamma \vdash \lstinline|not e| : \lstinline|Bool|\)}
+    \DisplayProof \\
+    % boolean ops
+    \AxiomC{\(\Gamma \vdash e_1 : \lstinline|Bool|\)}
+    \AxiomC{\(\Gamma \vdash e_2 : \lstinline|Bool|\)}
+    \BinaryInfC{\(\Gamma \vdash e_1 \land e_2 : \lstinline|Bool|\)}
+    \DisplayProof
+    \quad
+    \AxiomC{\(\Gamma \vdash e_1 : \lstinline|Bool|\)}
+    \AxiomC{\(\Gamma \vdash e_2 : \lstinline|Bool|\)}
+    \BinaryInfC{\(\Gamma \vdash e_1 \lor e_2 : \lstinline|Bool|\)}
+    \DisplayProof \\
+    %
+    % conditionals
+    \AxiomC{\(\Gamma \vdash e_1 : \lstinline|Bool|\)}
+    \AxiomC{\(\Gamma \vdash e_2 : \tau\)}
+    \AxiomC{\(\Gamma \vdash e_3 : \tau\)}
+    \TrinaryInfC{\(\Gamma \vdash if ~e_1~ then ~e_2~ else ~e_3 : \tau\)}
+    \DisplayProof \\
+    % pairs
+    \AxiomC{\(\Gamma \vdash e_1 : \tau_1\)}
+    \AxiomC{\(\Gamma \vdash e_2 : \tau_2\)}
+    \BinaryInfC{\(\Gamma \vdash (e_1, e_2) : (\tau_1, \tau_2)\)}
+    \DisplayProof \\
+    %
+    % projections
+    \AxiomC{\(\Gamma \vdash e : (\tau_1, \tau_2)\)}
+    \UnaryInfC{\(\Gamma \vdash fst(e) : \tau_1\)}
+    \DisplayProof
+    \quad
+    \AxiomC{\(\Gamma \vdash e : (\tau_1, \tau_2)\)}
+    \UnaryInfC{\(\Gamma \vdash snd(e) : \tau_2\)}
+    \DisplayProof \\
+    %
+    % function
+    \AxiomC{\(\Gamma \oplus \{x : \tau_1\} \vdash e : \tau_2\)}
+    \UnaryInfC{\(\Gamma \vdash x \Rightarrow e : \tau_1 \to \tau_2\)}
+    \DisplayProof \quad
+    %
+    % application
+    \AxiomC{\(\Gamma \vdash e_1 : \tau_1 \to \tau_2\)}
+    \AxiomC{\(\Gamma \vdash e_2 : \tau_1\)}
+    \BinaryInfC{\(\Gamma \vdash e_1 e_2 : \tau_2\)}
+    \DisplayProof
+    %
+  \end{gather*}
+
+  \pagebreak
+
+  \begin{enumerate}
+    \item Given the following type derivation with type variables \(\tau_1,
+    \ldots, \tau_5\), choose the correct options:
+    \begin{equation*}
+      \AxiomC{\((x, \tau_4) \in \Gamma\)}
+      \UnaryInfC{\(\Gamma \vdash x: \tau_4\)}
+      \UnaryInfC{\(\Gamma \vdash fst(x): \tau_3\)}
+      \AxiomC{\((x, \tau_4) \in \Gamma\)}
+      \UnaryInfC{\(\Gamma \vdash x: \tau_4\)}
+      \UnaryInfC{\(\Gamma \vdash snd(x): \tau_5\)}
+      \BinaryInfC{\(\Gamma \vdash fst(x)(snd(x)) : \tau_2\)}
+      \UnaryInfC{\(\Gamma' \vdash x \Rightarrow fst(x)(snd(x)): \tau_1\)}
+      \DisplayProof
+    \end{equation*}
+
+    \begin{enumerate}
+      \item There are no valid assignments to the type variables such that the
+      above derivation is valid.
+      \item In all valid derivations, \(\tau_2 = \tau_5\).
+      \item There are \emph{no} valid derivations where \(\tau_2 = \lstinline|Int|\).
+      \item In all valid derivations, \(\tau_4 = (\tau_3, \tau_5)\).
+      \item In all valid derivations, \(\tau_1 = \tau_4 \to \tau_2\).
+      \item There is a valid derivation where \(\tau_1 = \tau_2\).
+    \end{enumerate}
+    \item For each of the following pairs of terms and types, provide a valid
+    type derivation or briefly argue why the typing is incorrect:
+    \begin{enumerate}
+      \item \(x \Rightarrow x + 5\): \lstinline|Int| \(\to\) \lstinline|Int|
+      \item \(x \Rightarrow y \Rightarrow x + y\): \lstinline|Int| \(\to\)
+      \lstinline|Int| \(\to\) \lstinline|Int|
+      \item \(x \Rightarrow y \Rightarrow y(2) \times x\): \lstinline|Int| \(\to\)
+      \lstinline|Int| \(\to\) \lstinline|Int|
+      \item \(x \Rightarrow (x, x)\): \lstinline|Int| \(\to\) \lstinline|(Int, Int)|
+      \item \(x \Rightarrow y \Rightarrow if ~fst(x) ~then ~snd(x) ~else~ y\): \lstinline|(Bool, Int)| \(\to\) \lstinline|(Int, Int)| \(\to\) \lstinline|Int|
+      \item \(x \Rightarrow y \Rightarrow if ~y~ then~ (z \Rightarrow y) ~else~ x \): (\lstinline|Bool| \(\to\) \lstinline|Bool|) \(\to\) \lstinline|Bool| \(\to\) (\lstinline|Bool| \(\to\) \lstinline|Bool|)
+      \item \(x \Rightarrow y \Rightarrow if ~y~ then~ (z \Rightarrow y) ~else~ x \): (\lstinline|Int| \(\to\) \lstinline|Bool|) \(\to\) \lstinline|Bool| \(\to\) (\lstinline|Int| \(\to\) \lstinline|Bool|)
+    \end{enumerate}
+
+    \item Prove that there is \emph{no} valid type derivation for the term
+    \begin{equation*}
+      x \Rightarrow if~ fst(x) ~then~ snd(x)~ else~ x
+    \end{equation*}
+  \end{enumerate}
+
+  \begin{solution}
+
+    \begin{enumerate}
+      \item The correct statements are d and e. For the remaining:
+      \begin{itemize}
+        \item \textbf{a}: set \(\tau_2 = \lstinline|Int|\), \(\tau_5 =
+        \lstinline|Bool|\), \(\tau_3 = \tau_5 \to \tau_2\), \(\tau_4 = (\tau_3,
+        \tau_5)\), and \(\tau_1 = \tau_4 \to \tau_2\).
+        \item \textbf{b}: see (a).
+        \item \textbf{c}: see (a).
+        \item \textbf{f}: given \(\tau_1 = \tau_2\), we also know from the rule
+        for lambda abstraction that \(\tau_1 = \tau_4 \to \tau_2\), and hence
+        \(\tau_2 = \tau_4 \to \tau_2\) recursively, which is a contradiction.
+      \end{itemize}
+
+      \item For the given terms and types:
+      \begin{enumerate}
+        \item \(x \Rightarrow x + 5\): \lstinline|Int| \(\to\) \lstinline|Int|: \cmark
+        \begin{equation*}
+          \AxiomC{\(x : \lstinline|Int| \in \Gamma\)}
+          \UnaryInfC{\(\Gamma \vdash x : \lstinline|Int|\)}
+          \AxiomC{}
+          \UnaryInfC{\(\Gamma \vdash 5 : \lstinline|Int|\)}
+          \BinaryInfC{\(\Gamma \vdash x + 5 : \lstinline|Int|\)}
+          \UnaryInfC{\(\Gamma' \vdash x \Rightarrow x + 5 : \lstinline|Int| \to \lstinline|Int|\)}
+          \DisplayProof
+        \end{equation*}
+        \item \(x \Rightarrow y \Rightarrow x + y\): \lstinline|Int| \(\to\)
+        \lstinline|Int| \(\to\) \lstinline|Int|: \cmark
+        \item \(x \Rightarrow y \Rightarrow y(2) \times x\): \lstinline|Int|
+        \(\to\) \lstinline|Int| \(\to\) \lstinline|Int|: \xmark. If \(y\) has
+        type \lstinline|Int|, then \(y(2)\) cannot not well-typed, as the
+        function application rule is not applicable.
+        \item \(x \Rightarrow (x, x)\): \lstinline|Int| \(\to\) \lstinline|(Int, Int)|: \cmark
+        \item \(x \Rightarrow y \Rightarrow if ~fst(x) ~then ~snd(x) ~else~ y\):
+        \lstinline|(Bool, Int)| \(\to\) \lstinline|(Int, Int)| \(\to\)
+        \lstinline|Int|: \xmark. The type of the two branches of a conditional
+        must match, but here they are \lstinline|Int| and (\lstinline|Int|,
+        \lstinline|Int|) respectively.
+        \item \(x \Rightarrow y \Rightarrow if ~y~ then~ (z \Rightarrow y) ~else~ x \): (\lstinline|Bool| \(\to\) \lstinline|Bool|) \(\to\) \lstinline|Bool| \(\to\) (\lstinline|Bool| \(\to\) \lstinline|Bool|): \cmark
+        \begin{equation*}
+          \AxiomC{\((y, \lstinline|Bool|) \in \Gamma\)}
+          \UnaryInfC{\(\Gamma \vdash y : \lstinline|Bool|\)}
+          \AxiomC{\((x, \lstinline|Bool| \to \lstinline|Bool|) \in \Gamma\)}
+          \UnaryInfC{\(\Gamma \vdash x : \lstinline|Bool| \to \lstinline|Bool|\)}
+          \AxiomC{\((y, \lstinline|Bool|) \in \Gamma \oplus \{(z, \lstinline|Bool|)\}\)}
+          \UnaryInfC{\(\Gamma \oplus \{(z, \lstinline|Bool|)\} \vdash y : \lstinline|Bool|\)}
+          \UnaryInfC{\(\Gamma \vdash z \Rightarrow y : \lstinline|Bool| \to \lstinline|Bool|\)}
+          \TrinaryInfC{\(\Gamma \vdash if ~y~ then~ (z \Rightarrow y) ~else~ x : \lstinline|Bool| \to \lstinline|Bool|\)}
+          \UnaryInfC{\(\Gamma' \vdash y \Rightarrow if ~y~ then~ (z \Rightarrow y) ~else~ x : (\lstinline|Bool| \to \lstinline|Bool|) \to \lstinline|Bool| \to (\lstinline|Bool| \to \lstinline|Bool|)\)}
+          \UnaryInfC{\(\Gamma'' \vdash x \Rightarrow y \Rightarrow if ~y~ then~ (z \Rightarrow y) ~else~ x : (\lstinline|Bool| \to \lstinline|Bool|) \to \lstinline|Bool| \to (\lstinline|Bool| \to \lstinline|Bool|)\)}
+          \DisplayProof
+        \end{equation*}
+        Note that the choice of type of \(z\) (and of the argument of \(x\)) is
+        arbitrary. Hence, the next typing is also valid.
+        \item \(x \Rightarrow y \Rightarrow if ~y~ then~ (z \Rightarrow y) ~else~ x \): (\lstinline|Int| \(\to\) \lstinline|Bool|) \(\to\) \lstinline|Bool| \(\to\) (\lstinline|Int| \(\to\) \lstinline|Bool|): \cmark
+      \end{enumerate}
+      \item Non-existence of a valid type derivation for the term:
+      \begin{equation*}
+        t = x \Rightarrow if~ fst(x) ~then~ snd(x)~ else~ x
+      \end{equation*}
+      Assume that there is a valid type derivation for the term. We will attempt
+      to derive a contradiction. We use the fact that if there exists a type
+      derivation, every step must use one of the rules above, and that the types
+      assigned to each variable must be consistent across the derivation.
+      
+      First, \(t\) has a type derivation \emph{if and only if} \(t_1 = if ~
+      fst(x)~then~snd(x)~else~x\) has a type derivation, by using the function
+      abstraction rule. We will work with \(t_1\) directly. The function
+      abstraction rule here does not give us more information.
+
+      Any type derivation for \(t_1\) must end in the conditional rule. For this
+      rule to be applicable, we must have that the following are derivable:
+      \begin{enumerate}
+        \item \(\Gamma \vdash fst(x) : \lstinline|Bool|\)
+        \item \(\Gamma \vdash snd(x) : \tau\)
+        \item \(\Gamma \vdash x : \tau\)
+      \end{enumerate}
+      where the type variable \(\tau\) is also the type of \(t_1\).
+
+      By using the projection rule on (a) and (b), we learn that the type of
+      \(x\) must be \((\lstinline|Bool|, \tau_1)\) and \((\tau_2, \tau)\) for
+      two fresh variables \(\tau_1\) and \(\tau_2\) respectively. Matching the
+      two, as \(x\) may only have one type, we must have \(\tau_1 = \tau\),
+      \(\tau_2 = \lstinline|Bool|\), and thus the type of \(x\) is
+      \((\lstinline|Bool|, \tau)\).
+
+      However, from (c), we learn that the type of \(x\) is \(\tau\). It must be
+      the case that \(\tau = (\lstinline|Bool|, \tau)\). This is not possible
+      for any type \(\tau\), and we have a contradiction.
+
+      Hence, there is no valid type derivation for the term \(t\).
+    \end{enumerate}
+    
+  \end{solution}
+
+\end{exercise}

info/exercises/src/ex-04/main.tex

+\documentclass[a4paper]{article}
+
+\input{../macro}
+
+\ifdefined\ANSWERS
+  \if\ANSWERS1
+    \printanswers
+  \fi
+\fi
+
+\DeclareMathOperator{\half}{half}
+\newcommand{\num}{\ensuremath{\mathbf{num}}}
+
+\title{CS 320 \\ Computer Language Processing\\Exercise Set 4}
+\author{}
+\date{March 26, 2025}
+
+\begin{document}
+\maketitle
+
+% inductive relations
+%% grammars as inductive relations
+\input{ex/grammar}
+
+% operational semantics
+%% old ex 5 problems 1 and 2
+\input{ex/semantics}
+
+% type systems
+%% type derivations
+%% type system exte
+\input{ex/types}
+
+\end{document}

info/exercises/src/macro.tex

+% macro.tex
+% common to all exercises
+
+\usepackage[dvipsnames]{xcolor}
+\usepackage{amsmath, amssymb}
+\usepackage{xspace}
+\usepackage[colorlinks]{hyperref}
+\usepackage{tabularx}
+\usepackage{multicol}
+
+% for drawing
+\usepackage{tikz}
+\usetikzlibrary{automata, arrows, shapes, positioning}
+
+\usepackage{forest}
+
+\usepackage{bussproofs, bussproofs-extra}
+
+% for code
+\usepackage{listings}
+
+\definecolor{dkgreen}{rgb}{0,0.6,0}
+\definecolor{gray}{rgb}{0.5,0.5,0.5}
+\definecolor{mauve}{rgb}{0.58,0,0.82}
+
+\lstdefinestyle{scalaStyle}{
+  frame=tb,
+  language=scala,
+  aboveskip=3mm,
+  belowskip=3mm,
+  showstringspaces=false,
+  columns=flexible,
+  basicstyle=\small\ttfamily,
+  numbers=none,
+  numberstyle=\tiny\color{gray},
+  keywordstyle=\color{blue},
+  commentstyle=\color{dkgreen},
+  stringstyle=\color{mauve},
+  frame=none,
+  breaklines=true,
+  breakatwhitespace=true,
+  tabsize=2,
+}
+
+\lstset{style=scalaStyle}
+
+% lstinline in math mode
+% https://tex.stackexchange.com/a/127018
+
+\usepackage{letltxmacro}
+\newcommand*{\SavedLstInline}{}
+\LetLtxMacro\SavedLstInline\lstinline
+\DeclareRobustCommand*{\lstinline}{%
+  \ifmmode
+    \let\SavedBGroup\bgroup
+    \def\bgroup{%
+      \let\bgroup\SavedBGroup
+      \hbox\bgroup
+    }%
+  \fi
+  \SavedLstInline
+}
+
+% for exercises
+
+\newcounter{exercisenum}
+\newcommand{\theexercise}{\arabic{exercisenum}}
+
+\usepackage{marginnote}
+\newenvironment{exercise}[1]{\refstepcounter{exercisenum}\paragraph*{Exercise \theexercise}{\reversemarginpar\marginnote{#1}}}{}
+
+\usepackage{verbatim}
+
+\usepackage{ifthen}
+\newboolean{showanswers}
+\setboolean{showanswers}{false}
+\newcommand{\printanswers}{\setboolean{showanswers}{true}}
+\newcommand\suppress[1]{} 
+
+\newcommand\ite[3]{\ifthenelse{\boolean{#1}}%
+{#2\suppress{#3}}%
+{\suppress{#2}#3}}
+
+
+\newenvironment{solution}{%
+    \ite{showanswers}{\paragraph*{Solution}}{\expandafter\comment}
+}{%
+    \ite{showanswers}{\hfill\(\square\)}{\expandafter\endcomment}
+}
+
+\newcommand{\note}[1]{\hfill #1}
+
+\sloppy
+
+%% actual macros
+
+% meta
+\newcommand{\todo}[1]{\textcolor{red}{[Todo: #1]}}
+
+\newcommand{\fbowtie}{\mathrel \blacktriangleright \joinrel \mathrel \blacktriangleleft}
+
+\newcommand{\easy}{\ensuremath{\bigstar}\xspace}
+\newcommand{\hard}{\small\ensuremath{\fbowtie}\xspace}
+
+% real
+\newcommand{\naturals}{\mathbb{N}}
+\newcommand{\booleans}{\mathbb{B}}
+
+\usepackage{bussproofs}
+
+\usepackage{pifont}
+\newcommand{\cmark}{\ding{51}}
+\newcommand{\xmark}{\ding{55}}

info/grading.md

+# Grading Policy
+
+The grade is based on a midterm (30%) as well as team project work (70%).
+
+The project work is done in groups of 2-3 people (no individual groups; the goal is in part to learn how to work together).
+The work has many aspects: the implementation in [Scala](https://www.scala-lang.org/) of aspects of an interpreter and compiler (labs 1 to 5) and Lab 6, which is an open project. There will be no written exam at the _end_ of the semester and no exam in the exam period. Here are the weights of the milestones in the overall course grade:
+
+  * Midterm exam: 30% (see [the archive of past exams](past-exams/); note that we will have fewer multiple-choice questions this time)
+  * 10% Lab 1
+  * 10% Lab 2
+  * 10% Lab 3 (First team work statement to be sent afterwards)
+  * 10% Lab 4
+  * 10% Lab 5
+  * 20% Lab 6 (Compiler extension, customized, the final team work statement)
+
+After you receive your points for the submitted lab, you are allowed to discuss the lab with other group members and with teaching staff, so that you can correct it and continue to use your code in subsequent labs.
+
+Please note that, after the lab deadline, we reserve the right to ask you to explain any code that you submitted for the lab. You need to understand all the code submitted, regardless whether you or another group member wrote it. We will let you know in advance when you need to be present in the labs or exercises for such oral explanations on your laptop. Taking this into account, you are welcome to write and submit comments explaining what your code does.
+
+To monitor whether everyone is doing their share of work and help ensure that group members work together, we ask each student to submit via email their teamwork statement, twice during the semester: once right after Lab 3 is due, and once at the end of the semester. Please read carefully the [Teamwork Statements](teamwork.md) email instructions.
+
+For the final Lab 6, each group will need to do their own project (based on our suggestions or your own ideas that you check with the teaching staff). Each member of the group must present the project in a slot in one of the last two weeks of the semester and answer questions. The presentation part of of each person will be graded individually and includes answers to questions (a person not presenting will be given a 0 points for the presentation part of the Compiler extension lab). The final report on the project will need to handed in after the end of the semester but the students are encouraged to complete it during the semester as this is a continuous control course.

info/labs/.gitignore

+# sbt compilation output
+target/
+project/target
+# Compiler output and nodejs modules
+compiler/**/*.html
+compiler/**/*.wat
+compiler/**/*.wasm
+compiler/**/*.js
+compiler/node_modules
+compiler/package-lock.json
+# Latex output
+*.out
+*.aux
+*.log
+*.nav
+*.snm
+*.toc
+*.vrb
+# Vim
+*.swp
+# IntelliJ
+.idea
+# VSCode
+.vscode
+# tester
+tester/repos
+# Metals & Bloop
+.bsp/
+.bloop/
+.metals/
+metals.sbt

info/labs/amy-specification/AmySpec.md

+# Specification of the Amy Language
+
+**Computer Language Processing**  
+**LARA**  
+**Spring 2025**
+
+## 1. Introduction
+
+Welcome to the Amy project! This semester you will learn how to compile a simple functional Scala-like language from source files down to executable code. When your compiler is complete, it will be able to take Amy source (text) files as input and produce WebAssembly bytecode files. WebAssembly is a new format for portable bytecode which is meant to be run in browsers.
+
+This document is the specification of Amy. Its purpose is to help you clearly and unambiguously understand what an Amy program means, and to be the Amy language reference, along with the reference compiler. It does not deal with how you will actually implement the compiler; this will be described to you as assignments are released.
+
+**Note**: The language might change along the way, so check this specification before starting each lab, to make sure you have the latest version in mind.
+
+### 1.1 Features of Amy
+
+Let us demonstrate the basic features of Amy through some examples:
+
+#### 1.1.1 The factorial function
+
+```scala
+object Factorial
+
+def fact(i: Int(32)): Int(32) = {
+  if (i < 2) { 1 }
+  else { i * fact(i-1) }
+}
+
+end Factorial
+```
+
+Every program in Amy is contained in a module, also called `object`. A function is introduced with the keyword `def`, and all its parameters and result type must be explicitly typed. Amy supports conditional (or if-) expressions with obligatory brackets. Notice that conditionals are not statements, but return a value, in this case an `Int(32)`.
+
+In fact, there is no distinction between expressions and statements in Amy. Even expressions that are called only for their side-effects return a value of type `Unit`.
+
+The condition of an if-expression must be of type `Boolean` and its branches must have the same type, which is also the type of the whole expression.
+
+#### 1.1.2 Saying hello
+
+```scala
+object Hello
+  Std.printString("Hello " ++ "world!")
+end Hello
+```
+
+Amy supports compiling multiple modules together. To refer to functions (or other definitions) in another module, one must explicitly use a qualified name. There is no import statement like in Scala.
+
+In this example, we refer to the `printString` function in the `Std` module, which contains some builtin functions to interact with the user. The string we print is constructed by concatenating two smaller strings with the `++` operator.
+
+#### 1.1.3 Input, local variables and sequencing expressions
+
+```scala
+object ReadName
+  Std.printString("What is your name?");
+  val name: String = Std.readString();
+  Std.printString("Hello " ++ name)
+end ReadName
+```
+
+We can read input from the console with the `readX` functions provided in `Std`.
+
+We can define local variables with `val`, which must always be typed explicitly. The value of the variable is given after `=`, followed by a semicolon.
+
+We can sequence expressions with `;`. The value of the first expression is discarded, and the value of the second one is returned. Note that `;` is an operator and not a terminator: you are not allowed to put it at the end of a sequence of expressions.
+
+#### 1.1.4 Type definitions
+
+Except for the basic types, a user can define their own types in Amy. The user-definable types in Amy come from functional programming and are called algebraic data types. In this case, we define a type, `List`, and two constructors `Nil` and `Cons`, which we can call to construct values of type `List`.
+
+```scala
+object L
+  abstract class List
+  case class Nil() extends List
+  case class Cons(h: Int(32), t: List) extends List
+end L
+```
+
+#### 1.1.5 Constructing ADT values
+
+```scala
+def range(from: Int(32), to: Int(32)): List = {
+  if (to < from) { Nil() }
+  else {
+    Cons(from, range(from + 1, to))
+  }
+}
+```
+
+We can create a `List` by calling one of its two constructors like a function, as demonstrated in the `range` function.
+
+#### 1.1.6 Pattern matching
+
+```scala
+def length(l: List): Int(32) = {
+  l match {
+    case Nil() => 0
+    case Cons(h, t) => 1 + length(t)
+  }
+}
+```
+
+To use a list value in any meaningful way, we have to break it down, according to the constructor used to construct it. This is called pattern matching and is a powerful feature of functional programming.
+
+In `length` we pattern match against the input value `l`. Pattern matching will check if its argument matches the pattern of the first case, and if so will evaluate the corresponding expression. Otherwise it will continue with the second case etc. If no pattern matches, the program will exit with an error. If the constructor has arguments, as does `Cons` in this case, we can bind their values to fresh variables in the pattern, so we can use them in the case expression.
+
+#### 1.1.7 Wildcard patterns and errors
+
+The `error` keyword takes a string as argument, prints `Error:` and its argument on the screen, then exits the program immediately with an error code. In this function, we are trying to compute the head of a list, which should fail if the list is empty.
+
+Notice that in the second case, we don’t really care what the tail of the list is. Therefore, we use a wildcard pattern (`_`), which matches any value without binding it to a name.
+
+```scala
+def head(l: List): Int(32) = {
+  l match {
+    case Cons(h, _) => h
+    case Nil() => error("head(Nil)")
+  }
+}
+```
+
+### 1.2 Relation to Scala
+
+Amy, with mild syntactic variations, is designed to be as close to a simple subset of Scala as possible. However, it is not a perfect subset. You can easily come up with Amy programs that are not legal in Scala. However, many “reasonable” programs will be compilable with `scalac`, provided you provide an implementation of the Amy standard library along with your code. This should not be required however, as we are providing a reference implementation of Amy.
+
+---
+
+## 2. Syntax
+
+The syntax of Amy is given formally by the context-free grammar of Figure 1. Everything spelled in *italic* is a nonterminal symbol of the grammar, whereas the terminal symbols are spelled in monospace font. `*` is the Kleene star, `s+` stands for one or more repetitions of `s`, and `?` stands for optional presence of a symbol (zero or one repetitions). The square brackets `[]` are not symbols of the grammar; they merely group symbols together. Please note that the square brackets `[]` are still tokenized as they are reserved for future use.
+
+Before parsing an Amy program, the Amy lexer generates a sequence of terminal symbols (tokens) from the source files. Some non-terminal symbols mentioned, but not specified, in Figure 1 are also represented as a single token by the lexer. They are lexed according to the rules in Figure 2. In Figure 2, we denote the range between characters α and β (included) with `[α-β]`.
+
+The syntax in Figure 1 is an *overapproximation* of the real syntax of Amy. This means that it allows some programs that should not be allowed in Amy. To get the real syntax of Amy, there are some additional restrictions presented (among other things) in the following notes:
+
+- The reserved words of Amy are the following:  
+  `abstract`, `Boolean`, `case`, `class`, `def`, `else`, `error`, `extends`, `false`, `if`, `Int`, `match`, `object`, `end`, `String`, `true`, `Unit`, `val`, `_` (the wildcard pattern).  
+
+  Identifiers are not allowed to coincide with a reserved word.
+
+- The operators and language constructs of Amy have the following precedence, starting from the lowest:
+
+  1) `val`, `;`  
+  2) `if`, `match`  
+  3) `||`  
+  4) `&&`  
+  5) `==`  
+  6) `<`, `<=`  
+  7) `+`, `-`, `++`  
+  8) `*`, `/`, `%`  
+  9) Unary `-`, `!`  
+  10) `error`, calls, variables, literals, parenthesized expressions.
+
+  For example,  
+  `1 + 2 * 3` means `1 + (2 * 3)` and  
+  `1 + 2 match {...}` means `(1 + 2) match {...}`.
+
+  A little more complicated is the interaction between `;` and `val`:  
+  the definition part of the `val` extends only as little as the first semicolon, but then the variable defined is visible through any number of semicolons. Thus `(val x: Int(32) = y; z; x)` means `(val x: Int(32) = y; (z; x))` and not `(val x: Int(32) = (y; z); x)` or `((val x: Int(32) = y; z); x)` (i.e. `x` takes the value of `y` and is visible until the end of the expression).
+
+  All operators are left-associative. That means that within the same precedence category, the leftmost application of an operator takes precedence. An exception is the sequence operator, which for ease of the implementation (you will understand during parsing) can be considered right-associative (it is an associative operator so it does not really matter).
+
+- A `val` definition is not allowed directly in the value assigned by an enclosing `val` definition. E.g.  
+  `(val x: Int(32) = val y: Int(32) = 0; 1; 2)` is **not** allowed.  
+  On the other hand,  
+  `(val x: Int(32) = 0; val y: Int(32) = 1; 2)` is allowed.
+
+- It is not allowed to use a `val` as a (second) operand to an operator. E.g.  
+  `(1 + val x: Int(32) = 2; x)` is **not** allowed.
+
+- A unary operator is not allowed as a direct argument of another unary operator. E.g. `--x` is **not** allowed.
+
+- It is not allowed to use `match` as a first operand of any binary operator, except `;`. E.g. `(x match { ... } + 1)` is **not** allowed. On the other hand `(x match { ... }; x)` is allowed.
+
+- The syntax `[ Id . ]? Id` refers to an optionally qualified name, for example either `MyModule.foo` or `foo`. If the qualifier is included, the qualified name refers to a definition `foo` in another module `MyModule`; otherwise, `foo` should be defined in the current module. Since Amy does not have the import statement of Scala or Java, this is the only way to refer to definitions in other modules.
+
+- One line comments are introduced with `//`: `//This is a comment.` Everything until the end of the line is a comment and should be ignored by the lexer.
+
+- Multiline comments can be used as follows: `/*This is a comment */`. Everything between the delimiters is a comment, notably including newline characters and `/*`. **Nested comments are not allowed**.
+
+- Escaped characters are not recognised inside string literals. I.e. `"\n"` stands for a string literal which contains a *backslash* and an `n`.
+
+---
+
+```text
+Figure 1: Syntax of Amy
+
+Program ::= Module∗
+
+Module ::= object Id Definition∗ Expr? end Id
+Definition ::= AbstractClassDef | CaseClassDef | FunDef
+
+AbstractClassDef ::= abstract class Id
+CaseClassDef ::= case class Id ( Params ) extends Id
+
+FunDef ::= def Id ( Params ) : Type = { Expr }
+Params ::= ϵ | ParamDef [ , ParamDef ]∗
+
+ParamDef ::= Id : Type
+Type ::= Int ( 32 ) | String | Boolean | Unit | [ Id . ]? Id
+Expr ::= Id
+       | Literal
+       | Expr BinOp Expr
+       | UnaryOp Expr
+       | [ Id . ]? Id ( Args )
+       | Expr ; Expr
+       | val ParamDef = Expr ; Expr
+       | if ( Expr ) { Expr } else { Expr }
+       | Expr match { MatchCase+ }
+       | error ( Expr )
+       | ( Expr )
+
+Literal ::= true | false | ( )
+          | IntLiteral | StringLiteral
+
+BinOp ::= + | - | * | / | % | < | <=
+        | && | || | == | ++
+
+UnaryOp ::= - | !
+
+MatchCase ::= case Pattern => Expr
+Pattern ::= [ Id . ]? Id ( Patterns ) | Id | Literal | _
+Patterns ::= ϵ | Pattern [ , Pattern ]∗
+
+Args ::= ϵ | Expr [ , Expr ]∗
+```
+
+```text
+Figure 2: Lexical rules for Amy
+
+IntLiteral ::= Digit+
+
+Id ::= Alpha AlphaNum∗ (and not a reserved word)
+AlphaNum ::= Alpha | Digit | _
+Alpha ::= [a-z] | [A-Z]
+Digit ::= [0-9]
+
+StringLiteral ::= " StringChar∗ "
+StringChar ::= Any character except newline and "
+```
+
+---
+
+## 3. Semantics
+
+In this section we will give the semantics of Amy, i.e. we will systematically explain what an Amy program represents, as well as give the restrictions that a legal Amy program must obey. The discussion will be informal, except for the typing rules of Amy.
+
+### 3.1 Program Structure
+
+An Amy program consists of one or more source files. Each file contains a single module (object), which in turn consists of a series of type and function definitions, optionally followed by an expression. We will use the terms *object* and *module* interchangeably.
+
+### 3.2 Execution
+
+When an Amy program is executed, the expression at the end of each module, if present, is evaluated. The order of execution among modules is the same that the user gave when compiling or interpreting the program. Each module’s definitions are visible within the module automatically, and in all other modules provided a qualified name is used.
+
+### 3.3 Naming rules
+
+In this section, we will give the restrictions that a legal Amy program must obey with regard to naming or referring to entities defined in the program. Any program not following these restrictions should be rejected by the Amy name analyzer.
+
+- Amy is case-sensitive.
+- No two modules in the program can have the same name.
+- No two classes, constructors, and/or functions in the same module can have the same name.
+- No two parameters of the same function can have the same name.
+- No two local variables of the same function can have the same name if they are visible from one another. This includes binders in patterns of match-expressions. Variables that are not mutually visible can have the same name. E.g. the program  
+  `val x : Int(32) = 0; val x : Int(32) = 1; 2` is not legal, whereas  
+  `(val x : Int(32) = 0; 1); (val x : Int(32) = 1; 2)` is.
+- A local variable can have the same name as a parameter. In this case, the local variable definition shadows the parameter definition.
+- Every variable encountered in an expression has to refer to a function parameter or a local variable definition.
+- All case classes have to extend a class in the same module.
+- All function or constructor calls or type references have to refer to a function/constructor/type defined in the same module, or another module provided a qualified name is given. It is allowed to refer to a constructor/type/function before declaring it.
+- All calls to constructors and functions have to have the same number of arguments as the respective constructor/function definition.
+Below is the **Types and Classes** section (Section 3.4) reproduced more faithfully, in Markdown format:
+
+### 3.4 Types and Classes
+
+Every expression, function parameter, and class parameter in Amy has a type. Types catch some common programming errors by introducing typing restrictions. Programs that do not obey these restrictions are illegal and will be rejected by the Amy type checker.
+
+The built-in types of Amy are `Int(32)`, `String`, `Boolean` and `Unit`. `Int(32)` represents 32-bit signed integers. `String` is a sequence of characters. Strings have poor support in Amy: the only operations defined on them are concatenation and conversion to integer. In fact, not even equality is “properly” supported (see Section 3.5). `Boolean` values can take the values `true` and `false`. `Unit` represents a type with a single value, `()`. It is usually used as the result of a computation which is invoked for its side-effects only, for example, printing some output to the user. It corresponds to Java’s `void`.
+
+In addition to the built-in types, the programmer can define their own types. The sort of types that are definable in Amy are called **Algebraic Data Types (ADTs)** and come from the functional programming world, but they have also been successfully adopted in Scala.
+
+An ADT is a type along with several constructors that can create values of that type. For example, an ADT defining a list of integers in pseudo syntax may look like this:
+
+```scala
+type List = Nil() | Cons(Int(32), List)
+```
+
+which states that a `List` is either `Nil` (the empty list), or a `Cons` of an integer and another list. We will say that `Cons` has two fields of types `Int(32)` and `List`, whereas `Nil` has no fields. Inside the program, the only way to construct values of the `List` type is to call one of these constructors, e.g. `Nil()` or `Cons(1, Cons(2, Nil()))`. You can think of them as functions from their field types to the `List` type.
+
+Notice that in the above syntax, `Nil` and `Cons` are not types. More specifically, they are not *subtypes* of `List`: in fact, there is no subtyping in Amy. Only `List` is a type, and values such as `Nil()` or `Cons(1, Cons(2, Nil()))` have the type `List`.
+
+In Amy, we use Scala syntax to define ADTs. A type is defined with an abstract class and the constructors with case classes. The above definition in Amy would be:
+
+```scala
+abstract class List
+case class Nil() extends List
+case class Cons(h: Int(32), t: List) extends List
+```
+
+Notice that the names of the fields have no practical meaning, and we only use them to stay close to Scala.
+
+We will sometimes use the term **abstract class** for a type and **case class** for a type constructor.
+
+The main programming structure to manipulate class types is pattern matching. In Section 3.5 we define how pattern matching works.
+
+### 3.5 Typing Rules and Semantics of Expressions
+
+Each expression in Amy is associated with a typing rule, which constrains and connects its type and the types of its subexpressions. An Amy program is said to *typecheck* if:
+1) All its expressions obey their respective typing rules,
+2) The body of each function corresponds to its declared return type.
+
+A program that does not typecheck will be rejected by the compiler.
+
+In the following, we will informally give the typing rules and explain the semantics (meaning) of each type of expression in Amy. We will use function type notation for typing of the various operators. For example, `(A,B) => C` denotes that an operator takes arguments of types `A` and `B` and returns a value of type `C`.
+
+When talking about the semantics of an expression we will refer to a *context*. A context is a mapping from variables to the values that have been assigned to them.
+
+- **Literals** of Amy are expressions of the base types that are values, i.e. they cannot be evaluated further. The literals `true` and `false` have type `Boolean`. `()` (the unit literal) has type `Unit`. String literals have type `String` and integer literals have type `Int(32)`.
+
+- A **variable** has the type of the corresponding definition (function parameter or local variable definition). Its value is the value assigned to it in the current context.
+
+- `+`, `-`, `*`, `/` and `%` have type `(Int(32), Int(32)) => Int(32)`, and are the usual integer operators.
+
+- Unary `-` has type `(Int(32)) => Int(32)` and is the integer negation.
+
+- `<` and `<=` have type `(Int(32), Int(32)) => Boolean` and are the usual arithmetic comparison operators.
+
+- `&&` and `||` have type `(Boolean, Boolean) => Boolean` and are the boolean conjunction and disjunction. Notice that these operators are short-circuiting. This means that the second argument does not get evaluated if the result is known after computing the first one. For example, `true || error("")` will yield `true` and not result in an error, whereas `false || error("")` will result in an error in the program.
+
+- `!` has type `(Boolean) => Boolean` and is the boolean negation.
+
+- `++` has type `(String, String) => String` and is the string concatenation operator.
+
+- `==` is the equality operator. It has type `(A,A) => Boolean` for every type `A`. 
+  - Equality for values of the `Int(32)`, `Boolean` and `Unit` types is defined as value equality, i.e. two values are equal if they have the same representation. E.g. `0 == 0`, `() == ()` and `(1 + 2) == 3`.
+  - Equality for the reference types `String` and all user-defined types is defined as reference equality, i.e. two values are equal only if they refer to the same object. I.e. `"" == ""`, `"a" ++ "b" == "ab"` and `Nil() == Nil()` all evaluate to `false`, whereas  
+    `(val s = "Hello"; s == s)` evaluates to `true`.
+
+- `error()` has type `(String) => A` for any type `A`, i.e. `error` is always acceptable, regardless of the expected type. When a program encounters `error`, it needs to print something like `Error: <msg>`, where `<msg>` is its evaluated argument, and then exit immediately.
+
+- `if(..) {..} else {..}` has type `(Boolean, A, A) => A` for any type `A`, and has the following meaning: First, evaluate the condition of `if`. If it evaluates to `true`, evaluate and return the then-branch; otherwise, evaluate and return the else-branch. Notice that the value that is not taken is not evaluated.
+
+- `;` is the sequence operator. It has type `(A, B) => B` for any types `A` and `B`. Notice that the first expression has to be well typed, although its precise type does not matter. `;` evaluates and discards its first argument (which we will usually invoke just for its side-effects) and then evaluates and returns its second argument.
+
+- `val n = e; b` defines a local variable with name `n` and adds it to the context, mapped to the value of `e`. It is visible in `b` but not in `e`. `n` has to obey the name restrictions described in Section 3.3.
+
+- An expression `f(..)` or `m.f(..)` denotes either a function call or an invocation of a type constructor. `f` has to be the name of a function/constructor defined in the program. The types of the real arguments of the function/constructor invocation have to match the corresponding types of the formal arguments in the definition of the function/constructor. The type of a function/constructor call is the return type of the function, or the parent type of the constructor respectively.
+
+  Evaluating a function call means evaluating its body in a new context, containing the function’s formal arguments mapped to the values of the real arguments provided at the function call. Evaluating a call to a constructor means generating and returning a fresh object containing (a reference to) the constructor and the arguments passed to it. Notice that an invocation of a type constructor on values is itself a value, i.e. cannot be evaluated further. It corresponds to literals of the other types.
+
+- `match` is the pattern-matching construct of Amy. It corresponds to Scala’s pattern matching. Java programmers can think of it as a generalized `switch`-statement. `match` is the only way to access the structure of a value of a class type. It also happens to be the most complicated structure of Amy.
+
+  **Terminology:** To explain how the match-expression works, let us first establish some terminology. A match expression has a *scrutinee* (the first operand, which gets pattern matched on), and a number of *match cases* (or simply cases). A case is introduced with the keyword `case`, followed by the (case) pattern, then the symbol `=>` and finally an expression, which we will call the *case expression*.
+
+  As seen in Section 2, a pattern comes in four different forms, which in the grammar are denoted as:
+  1) `Id(Patterns)` — **case class pattern**  
+  2) `Id` — **identifier pattern**  
+  3) `Literal` — **literal pattern**  
+  4) `_` — **wildcard pattern**  
+
+  The identifier at the beginning of **case class pattern** is called the constructor of the pattern, and its arguments are called its *subpatterns*.
+
+  **Typing rules:** For the match-expression to typecheck, two conditions have to hold:
+  1) All its case expressions have the same type, which is also the type of the whole match expression.  
+  2) All case patterns have to follow the type of the scrutinee. For a pattern to follow a type means the following, according to its form:
+     - Each literal pattern follows exactly the type of its literal.
+     - Wildcard and identifier patterns follow any type.
+     - A case class pattern follows only the resulting type of its constructor, if and only if all its subpatterns follow the types of the respective fields of the constructor.
+
+  For example, `Nil() match { case Cons(_, t) => () }` typechecks, whereas `Nil() match { case 0 => () }` does not.
+
+  **Semantics:** The semantics of pattern matching are as follows: First, the scrutinee is evaluated, then cases are scanned one by one until one is found whose pattern matches the scrutinee value. If such case is found, its case expression is evaluated, after adding to the environment the variables bound in the case pattern (see below). The value produced in this way is returned as the value of the match-expression. If none is found, the program terminates with an error.
+
+  We say that a pattern *matches* a value when the following holds:
+  - A wildcard pattern (`_`) or an identifier pattern (e.g. `x`) match any value. In the second case, `x` is bound to that value when evaluating the case expression.
+  - A literal pattern matches exactly the value of its literal. Notice that string literals are compared by reference, so they can never match.
+  - A case class pattern `case C(..)` matches a value `v`, if and only if `v` has been constructed with the same constructor `C` and every subpattern of the pattern matches the corresponding field of `v`. Notice that we have to recursively bind identifiers in subpatterns.
+
+- **Parentheses** `(e)` can be used freely around an expression `e`, mainly to override operator precedence or to make the program more readable. `(e)` is equivalent to `e`.
+
+- When evaluating an expression composed of sub-expressions (e.g. `f(a,b)`), the different sub-expressions are evaluated in left-to-right order (i.e. in the previous example, the sub-expressions would be evaluated in the following order: `f` then `a` then `b`). Function calls are done using the *call by value* strategy.
+
+### 3.6 Formal discussion of types
+
+In this section, we give a formal (i.e. mathematically robust) description of the Amy typing rules. A typing rule will be given as:
+
+```text
+   Rule Name
+   P1  ...  Pn
+   -----------
+        C
+```
+
+where `Pi` are the rule premises and `C` is the rule conclusion. A typing rule means that the conclusion is true under the premises.
+
+Conclusions and most premises will be type judgements in an environment. A type judgement `Γ ⊢ e : T` means that an expression (or pattern) `e` has type `T` in environment `Γ`. Environments `Γ` are mappings from variables to types and will be written as `Γ = v1 : T1, . . . , vn : Tn`. We can add a new pair to an environment `Γ` by writing `Γ, vn+1 : Tn+1`. We will also sometimes write a type judgement of the form `Γ ⊢ p`. This means that `p` typechecks, but we don’t assign a type to it. Type checking will try to typecheck a program under the initial environment, and reject the program if it fails to do so.
+
+The initial environment `Γ0(p)` of a program `p` is one that contains the types of all functions and constructors in `p`, where a constructor is treated as a function from its fields to its parent type (see Section 3.4). The initial environment is used to kickstart typechecking at the function definition level.
+
+Below are the typing rules for expressions, patterns, functions, and programs:
+
+```text
+Figure 3: Typing rules for expressions
+
+Variable
+v : T ∈ Γ
+-----------
+Γ ⊢ v : T
+
+Int Literal
+i is an integer literal
+------------------------
+Γ ⊢ i : Int(32)
+
+String Literal
+s is a string literal
+----------------------
+Γ ⊢ s : String
+
+Unit
+--------------
+Γ ⊢ () : Unit
+
+Boolean Literal
+b ∈ {true, false}
+------------------
+Γ ⊢ b : Boolean
+
+Arith. Bin. Operators
+Γ ⊢ e1 : Int(32)   Γ ⊢ e2 : Int(32)
+op ∈ {+, -, *, /, %}
+--------------------------------------
+Γ ⊢ e1 op e2 : Int(32)
+
+Arith. Comp. Operators
+Γ ⊢ e1 : Int(32)   Γ ⊢ e2 : Int(32)
+op ∈ {<, <=}
+-------------------------------------
+Γ ⊢ e1 op e2 : Boolean
+
+Arith. Negation
+Γ ⊢ e : Int(32)
+-----------------
+Γ ⊢ -e : Int(32)
+
+Boolean Bin. Operators
+Γ ⊢ e1 : Boolean   Γ ⊢ e2 : Boolean
+op ∈ {&&, ||}
+-------------------------------------
+Γ ⊢ e1 op e2 : Boolean
+
+Boolean Negation
+Γ ⊢ e : Boolean
+------------------
+Γ ⊢ !e : Boolean
+
+String Concatenation
+Γ ⊢ e1 : String   Γ ⊢ e2 : String
+-----------------------------------
+Γ ⊢ e1 ++ e2 : String
+
+Equality
+Γ ⊢ e1 : T   Γ ⊢ e2 : T
+-------------------------
+Γ ⊢ e1 == e2 : Boolean
+
+Sequence
+Γ ⊢ e1 : T1   Γ ⊢ e2 : T2
+---------------------------
+Γ ⊢ e1 ; e2 : T2
+
+Local Variable Definition
+Γ ⊢ e1 : T1   Γ, n : T1 ⊢ e2 : T2
+-----------------------------------
+Γ ⊢ val n : T1 = e1 ; e2 : T2
+
+Function/Class Constructor Invocation
+Γ ⊢ e1 : T1 ... Γ ⊢ en : Tn
+Γ ⊢ f : (T1, ... , Tn) ⇒ T
+--------------------------------
+Γ ⊢ f(e1, ... , en) : T
+
+If-Then-Else
+Γ ⊢ e1 : Boolean   Γ ⊢ e2 : T   Γ ⊢ e3 : T
+--------------------------------------------
+Γ ⊢ if (e1) {e2} else {e3} : T
+
+Error
+Γ ⊢ e : String
+---------------
+Γ ⊢ error(e) : T
+
+Pattern Matching
+Γ ⊢ e : Ts
+∀i ∈ [1, n]. Γ ⊢ pi : Ts
+∀i ∈ [1, n]. Γ, bindings(pi) ⊢ ei : Tc
+-------------------------------------------------------------
+Γ ⊢ e match { case p1 => e1 ... case pn => en } : Tc
+```
+
+```text
+Figure 4: Typing rules for patterns, functions and programs
+
+Wildcard Pattern
+-------------
+Γ ⊢ _ : T
+
+Identifier Pattern
+--------------
+Γ ⊢ v : T
+
+Case Class Pattern
+Γ ⊢ p1 : T1 ... Γ ⊢ pn : Tn
+Γ ⊢ C : (T1, ... , Tn) ⇒ T
+----------------------------------
+Γ ⊢ C(p1, ... , pn) : T
+
+Function Definition
+Γ, v1 : T1, ... , vn : Tn ⊢ e : T
+---------------------------------------------
+Γ ⊢ def f(v1 : T1, ... , vn : Tn): T = { e }
+
+Program
+∀f ∈ p. Γ0(p) ⊢ f
+--------------------
+⊢ p
+```
+
+---
+
+## 4. The standard library of Amy
+
+Amy comes with a library of predefined functions, which are accessible in the `Std` object. Some of these functions implement functionalities that are not expressible in Amy, e.g. printing to the standard output. These built-in functions are implemented in JavaScript and WebAssembly in case of compilation, and in Scala in the interpreter. Built-in functions have stub implementations in the Amy `Std` module for purposes of name analysis and type checking.
+
+The Amy compiler will not automatically include `Std` to the input files. If you want them included, you have to provide them manually.
+
+The signature of the `Std` module is the following:
+
+```scala
+object Std
+
+// Output
+def printString(s: String): Unit = ...
+def printInt(i: Int(32)): Unit = ...
+def printBoolean(b: Boolean): Unit = ...
+
+// Input
+def readString(): String = ...
+def readInt(): Int(32) = ...
+
+// Conversions
+def intToString(i: Int(32)): String = ...
+def digitToString(i: Int(32)): String = ...
+def booleanToString(b: Boolean): String = ...
+
+end Std
+```

info/labs/lab01/amyi.sh

+#!/bin/bash
+# Script similar in structure to amytc.sh, see explanations there.
+if [ $# -eq 0 ]; then
+    echo "Usage: amyi Prog1.amy Prog2.amy ... ProgN.amy"
+    exit 1
+fi
+echo Intepreting: $*
+AMYJAR=target/scala-3.5.2/amyc-assembly-1.7.jar
+if test -r "${AMYJAR}"; then
+    echo "Reusing existing jar: ${AMYJAR}"
+else
+    sbt assembly
+fi
+java -jar ${AMYJAR} --interpret library/Std.amy library/Option.amy library/List.amy $*

info/labs/lab01/amytc.sh

+#!/bin/bash
+# The above line tells the operating system to use the 
+# bash shell interpreter to execute the commands in this file.
+# the hash sign, '#' means that characters following it are comments
+if [ $# -eq 0 ]; then # script is called with 0 parameters
+    # output short usage instructions on the command line
+    echo "Usage: amytc.sh Prog1.amy Prog2.amy ... ProgN.amy"
+    echo "Example invocation:"
+    echo "./amytc.sh examples/Arithmetic.amy"
+
+    echo "Example output:"
+    echo "  Type checking: examples/Arithmetic.amy"
+    echo "  Reusing existing jar: target/scala-3.5.2/amyc-assembly-1.7.jar"
+    echo "  Type checking successful!"
+    # Now, terminate the script with an error exit code 1:
+    exit 1
+fi
+# print progress message. $* denotes all arguments
+echo Type checking: $*
+# jar file is a zip file containing .class files of our interpreter/compiler
+# sbt generates the file in this particular place.
+AMYJAR=target/scala-3.5.2/amyc-assembly-1.7.jar
+# You can copy this file into test.zip and run unzip test.zip to see inside
+if test -r "${AMYJAR}"; then
+    # jar file exists, so we just reuse it
+    echo "Reusing existing jar: ${AMYJAR}"
+    # Note that we do not check if scala sources changed!
+    # Hence, our jar file can be old
+else
+    # If there is no jar file, we invoke `sbt assembly` to create it
+    sbt assembly
+fi
+# We should have the jar file now, so we invoke it
+# java starts the Java Virtual Machine. 
+# Here, it will unpack the jar file and find META-INF/MANIFEST.MF file
+# which specifies the main class of the jar file (entry point).
+# java will execute `public static void main` method of that class.
+java -jar ${AMYJAR} --type-check library/Std.amy library/Option.amy library/List.amy $*
+# Here, we ask amy to only type check the give files. 
+# We always provide standard library files as well as 
+# the explicitly files explicit given to the script (denoted $*)

info/labs/lab01/build.sbt

+version := "1.7"
+organization := "ch.epfl.lara"
+scalaVersion := "3.5.2"
+assembly / test := {}
+name := "amyc"
+
+Compile / scalaSource := baseDirectory.value / "src"
+scalacOptions ++= Seq("-feature")
+
+Test / scalaSource := baseDirectory.value / "test" / "scala"
+Test / parallelExecution := false
+libraryDependencies += "com.novocode" % "junit-interface" % "0.11" % "test"
+libraryDependencies += "org.apache.commons" % "commons-lang3" % "3.4" % "test"
+testOptions += Tests.Argument(TestFrameworks.JUnit, "-v")
+
+

info/labs/lab01/examples/Arithmetic.amy

+object Arithmetic
+  def pow(b: Int(32), e: Int(32)): Int(32) = {
+    if (e == 0) { 1 }
+    else {
+      if (e % 2 == 0) {
+        val rec: Int(32) = pow(b, e/2);
+        rec * rec
+      } else {
+        b * pow(b, e - 1)
+      }
+    }
+  }
+
+  def gcd(a: Int(32), b: Int(32)): Int(32) = {
+    if (a == 0 || b == 0) {
+      a + b
+    } else {
+      if (a < b) {
+        gcd(a, b % a)
+      } else {
+        gcd(a % b, b)
+      }
+    }
+  }
+
+  Std.printInt(pow(0, 10));
+  Std.printInt(pow(1, 5));
+  Std.printInt(pow(2, 10));
+  Std.printInt(pow(3, 3));
+  Std.printInt(gcd(0, 10));
+  Std.printInt(gcd(17, 99)); // 1
+  Std.printInt(gcd(16, 46)); // 2
+  Std.printInt(gcd(222, 888)) // 222
+end Arithmetic

info/labs/lab01/examples/Factorial.amy

+object Factorial
+  def fact(i: Int(32)): Int(32) = {
+    if (i < 2) { 1 }
+    else { 
+      val rec: Int(32) = fact(i-1);
+      i * rec
+    }
+  }
+
+  Std.printString("5! = "  ++ Std.intToString(fact(5)));
+  Std.printString("10! = " ++ Std.intToString(fact(10)))
+end Factorial

info/labs/lab01/examples/Hanoi.amy

+object Hanoi
+	
+  def solve(n : Int(32)) : Int(32) = {
+    if (n < 1) { 
+      error("can't solve Hanoi for less than 1 plate")
+    } else {
+      if (n == 1) {
+        1
+      } else {
+        2 * solve(n - 1) + 1
+      }
+    }
+  }
+
+  Std.printString("Hanoi for 4 plates: " ++ Std.intToString(solve(4)))
+end Hanoi
\ No newline at end of file

info/labs/lab01/examples/Hello.amy

+object Hello
+  Std.printString("Hello " ++ "world!")
+end Hello

info/labs/lab01/examples/HelloInt.amy

+object HelloInt
+  Std.printString("What is your name?");
+  val name: String = Std.readString();
+  Std.printString("Hello " ++ name ++ "! And how old are you?");
+  val age: Int(32) = Std.readInt();
+  Std.printString(Std.intToString(age) ++ " years old then.")
+end HelloInt

info/labs/lab01/examples/Printing.amy

+object Printing
+  Std.printInt(0); Std.printInt(-222); Std.printInt(42);
+  Std.printBoolean(true); Std.printBoolean(false);
+  Std.printString(Std.digitToString(0));
+  Std.printString(Std.digitToString(5));
+  Std.printString(Std.digitToString(9));
+  Std.printString(Std.intToString(0));
+  Std.printString(Std.intToString(-111));
+  Std.printString(Std.intToString(22));
+  Std.printString("Hello " ++ "world!");
+  Std.printString("" ++ "")
+end Printing

info/labs/lab01/examples/TestLists.amy

+object TestLists 
+  val l: L.List = L.Cons(5, L.Cons(-5, L.Cons(-1, L.Cons(0, L.Cons(10, L.Nil())))));
+  Std.printString(L.toString(L.concat(L.Cons(1, L.Cons(2, L.Nil())), L.Cons(3, L.Nil()))));
+  Std.printInt(L.sum(l));
+  Std.printString(L.toString(L.mergeSort(l)))
+end TestLists

info/labs/lab01/lab01-README.md

+# Amy Lab 01: Interpreter
+
+Below you will find the instructions for the first lab assignment in which you will get to know and implement an interpreter for the Amy language.
+
+## Logistics
+
+As a reminder, the labs are done in groups of 2-3, please register on Moodle if not already done.
+
+We advice you to create a private git repository to track your work and collaborate.
+
+The labs are graded through Moodle assignments, similarly to Software Construction (CS-214) that you might have taken. You will have to submit your `.scala` files on Moodle and you will receive automatically a grade and feedback. You submit as many times as you want, only the last submission will be taken into account. The tests are the same as the ones you will receive for each lab, we do not use any hidden tests.
+
+For this first lab, you can download the initial project scaffold from this folder.
+
+## Part 1: Your first Amy programs
+
+Write two example Amy programs each make sure that they typecheck (see [Type check examples](#type-check-examples)). Put them under `/examples`. Please be creative when writing your programs: they should be nontrivial and not reproduce the functionality of the examples in the `/library` and `/examples` directories of the repository. Of course you are welcome to browse these directories for inspiration.
+
+Remember that you will use these programs in the remaining of the semester to test your compiler, so don't make them too trivial! Try to test many features of the language.
+
+If you have questions about how a feature of Amy works, you can always look at the [Amy Specification](../amy-specification/AmySpec.md). It's a good idea to keep a local copy of this document handy -- it will be your reference for whenever you are asked to implement an aspect of the Amy language throughout this semester.
+
+### Type check examples
+
+You can use the provided Frontend to type check your programs. To do so, run the provided bash script:
+
+```bash.sh
+./amytc.sh examples/your_program.amy
+```
+
+This will run the compiler frontend up to type checking and report either `Type checking successful!` or an error message. If you get an error message, you should fix the error before moving on to the next step.
+
+Please examine the bash scipt amytc.sh and its comments in your editor to understand how it works. Do not modify it.
+
+#### Troubleshooting
+
+- Your project must compile before you call the `amytc.sh` script.
+- If you get unexpected errors or behaviour, try to delete the `target/scala-3.5.2/amyc-assembly-1.7.jar` and retry.
+
+## Part 2: An Interpreter for Amy
+
+The main task of the first lab is to write an interpreter for Amy.
+
+### Interpreters
+
+The way to execute programs you have mostly seen so far is compilation to some kind of low-level code (bytecode for a virtual machine such as Java's; native binary code in case of languages such as C). An alternative way to execute programs is interpretation. According to Wikipedia, "an interpreter is a computer program that directly executes, i.e. performs, instructions written in a programming or scripting language, without previously compiling them into a machine language program". In other words, your interpreter is supposed to directly look at the code and *interpret* its meaning. For example, when encountering a call to the 'printString' function, your interpreter should print its argument on the standard output. This is the way Python is executing your code.
+
+### The general structure of the Interpreter
+
+The skeleton of the assignment is provided by us as an `sbt` project. See the [Implementation skeleton](#implementation-skeleton) section for more details.
+
+You will modify the `Interpreter.scala` file.
+
+In `Main.scala` you find the main method which is the entry point to your program. After processing the command line arguments of the interpreter, the main method creates a Pipeline, which contains the different stages of the compiler which you will implement in the future labs. The Pipeline will first call the Amy frontend, which will parse the source program into an abstract syntax tree (AST) and check it for correctness according to the [Amy Specification](../amy-specification/AmySpec.md), and then passes the result to the Interpreter.
+
+The AST abstracts away uninteresting things of the program (e.g. parentheses, whitespace, operator precedence...) and keeps the essential structure of the program. It describes the structure of programs recursively. For example, here you have the description of a module in Amy:
+
+`Module ::= **object** Id Definition* Expr? **end** Id`
+
+and in the implementation we find a class:
+
+`case class ModuleDef(name: Identifier, defs: List[ClassOrFunDef], optExpr: Option[Expr]) extends Definition`
+
+A comparison of the implementation of ASTs in Java (as shown in the book) and Scala is instructive.
+
+You can find the source code of the AST in the [TreeModule.scala](./src/amyc/ast/TreeModule.scala).
+
+### The Interpreter class
+
+Now let's delve into `Interpreter.scala`. This file currently only contains a partial implementation, and it is your task to complete it! The entrypoint into the interpreter is `interpret`, which takes an expression as input and executes its meaning. The main loop at the end of the class will just take the modules in order and interpret their expression, if present.
+
+`interpret` returns a `Value`, which is a type that represents a value that an Amy expression can produce. Value is inherited by classes which represent the different types of values present in Amy (`Int(32)`, `Booleans`, `Unit`, `String` and ADT values). `Value` has convenience methods to cast to `Int(32)`, `Boolean` and `String` (`as*`). Remember we can always call these methods safely when we know the types of an expression (e.g. the operands of an addition), since we know that the program type-checks.
+
+`interpret` takes an additional implicit parameter as an argument, which is a mapping from variables to values (in the interpreted language). In Scala, when an implicit parameter is expected, the compiler will look in the scope for some binding of the correct type and pass it automatically. This way we do not have to pass the same mapping over and over to all recursive calls to `interpret`. Be aware, however, that there are some cases when you need to change the `locals` parameter! Think carefully about when you have to do so.
+
+A few final notes:
+
+- You can print program output straight to the console.
+- You can assume the input programs are valid. This is guaranteed by the Amy frontend.
+- To find constructors and functions in the program, you have to search in the `SymbolTable` passed along with the program. To do so, use the three helper methods provided in the interpreter:
+  - `isConstrutor` will return whether the `Identifier` argument is a type constructor in the program
+  - `findFunctionOwner` will return the module which contains the given `Identifier`, which has to be a function in the program. E.g. if you give it the `printInt` function of the `Std` module, you will get the string `"Std"`.
+  - `findFunction` will return the function definition given a pair of Strings representing the module containing the function, and the function name. The return value is of type `FunDef` (see [the AST definitions](./src/amyc/ast/TreeModule.scala)).
+- When comparing Strings by reference, compare the two `StringValue`s directly and not the underlying Strings. The reason is that the JVM may return true when comparing Strings by equality when it is not expected (it has to do with JVM constant pools).
+- Some functions contained in the `Std` module are built-in in the language, i.e. they are hard-coded in the interpreter because they cannot be implemented in Amy otherwise. An example of a built-in function is `printString`. When you implement the interpreter for function calls, you should first check if the function is built-in, and if so, use the implementation provided in the `builtIns` map in the interpreter.
+- When a program fails (e.g. due to a call to `error` or a match fail), you should call the dedicated method in the Context: `ctx.reporter.fatal`.
+
+### Implementation skeleton
+
+You can get the project scaffold from [this folder](.).
+
+- `src/amyc/interpreter/Interpreter.scala` contains the partially implemented interpreter
+- `src/amyc/Main.scala` contains the `main` method which runs the interpreter on the input files
+- The `library` directory contains library definitions you can use in your Amy programs.
+- The `examples` directory contains some example programs on which you can try your implementation. Remember that most of them also use library files from `/library`. This should also contain the programs you wrote in Part 1.
+- `lib/amy-frontend-1.7.jar` contains the frontend of the compiler as a library, allowing you directly work with type-checked ASTs of input programs. You need this to be able to extract the AST from your source code to interpret it, as you did not implement this part of the compiler yet. This is also what allowed you to type check programs in part 1. **Note**: You are only allowed to use this binary code to link against your interpreter.
+
+You will have to complete the interpreter by implementing the missing methods (marked with the placeholder `???`).
+
+### Testing
+
+When you are done, use sbt to try some of your programs from Part 1:
+
+```bash
+  $ sbt
+  > run library/Std.amy examples/Hello.amy
+  Hello world!
+```
+
+You can also run your interpreter with the `amyi.sh` script in a similar way as you did with the type checker:
+
+```bash
+  $ ./amyi.sh examples/Hello.amy
+  Hello world!
+```
+
+**Note**: if you use this method, you have to delete `target/scala-3.5.2/amyc-assembly-1.7.jar` before running the script when you modified your interpreter. Otherwise, the script will reuse the previously compiled version of the interpreter and your new modifications would not be taken into account. Therefore this method is more recommended for testing multiple amy programs, rather than testing your interpreter while you are developing it.
+
+There is also testing infrastructure under `/test`. To add your own tests, you have to add your testcases under `/test/resources/interpreter/passing`
+and the expected output under `/test/resources/interpreter/outputs`.
+Then, you have to add the name of the new test in `InterpreterTests`, similarly to the examples given.
+To allow a test to also use the standard library (e.g., `Std.printString`), you can copy `Std.scala` from `library/Std.scala` to `/test/resources/interpreter/passing`.
+
+For example, to add a test that expects only "Hello world" to be printed, you can add `/test/resources/interpreter/passing/Hello.amy` containing:
+
+```scala
+object Hello 
+  Std.printString("Hello world") 
+end Hello
+```
+
+and `/test/resources/interpreter/outputs/Hello.txt` containing
+
+```text
+Hello world
+
+```
+
+(with a newline in the end!).
+
+You will also have to add a line to `/test/scala/amyc/test/InterpreterTests.scala`:  `@Test def testHello = shouldOutput(List("Std", "Hello"), "Hello")`. This will pass both files `Std.amy` and `Hello.amy` as inputs of the test. When you now run `test` from sbt, you should see the additional test case (called `testHello`).
+
+The tests provided originally in `test/` are the ones used to grade your work on Moodle. Please note that the grade returned by the grader on Moodle is what you will get for the lab. Therefore you should submit regularly on Moodle to validate your progress. Also, if tests pass locally but not on the grader, the grader is the one that counts so submit your work regularly and check the feedback in case of discrepancies.
+
+### Deliverables
+
+You should submit the following files on Moodle:
+
+- `Interpreter.scala` with your implementation of the interpreter
+
+Deadline: **07.03.2025 23:59:59**
+
+#### Related documentation
+
+- End of Chapter 1 in the Tiger Book presents a similar problem for another mini-language. A comparison of the implementation of ASTs in Java (as shown in the book) and Scala is instructive.