small-step-lab.ipynb

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    "Before you turn this problem in, make sure everything runs as expected. First, **restart the kernel** (in the menubar, select Kernel $\\rightarrow$ Restart) and then **run all cells** (in the menubar, select Cell $\\rightarrow$ Run All).\n",
    "\n",
    "Make sure you fill in any place that says `???`, `YOUR CODE HERE`, \"???\", \"YOUR ANSWER HERE\", as well as your name and collaborators below:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6d1f16a1",
   "metadata": {},
   "outputs": [],
   "source": [
    "val NAME = \"\"\n",
    "val COLLABORATORS = \"\""
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8fd4a79f",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
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   "id": "fa976307-2ff6-4380-a127-3682ac3452ed",
   "metadata": {},
   "source": [
    "$\\newcommand{\\TirName}[1]{\\text{#1}}\n",
    "\\newcommand{\\inferrule}[3][]{\n",
    "  \\let\\and\\qquad\n",
    "  \\begin{array}{@{}l@{}}\n",
    "  \\TirName{#1}\n",
    "  \\\\\n",
    "  \\displaystyle\n",
    "  \\frac{#2}{#3}\n",
    "  \\end{array}\n",
    "}\n",
    "\\newcommand{\\infer}[3][]{\\inferrule[#1]{#2}{#3}}\n",
    "$\n",
    "\n",
    "# Lab: Small-Step Operational Semantics\n",
    "\n",
    "<!-- 3 Expressions -->\n",
    "\n",
    "<!-- 4 Binding and Scope -->\n",
    "\n",
    "<!-- 8 Recursion -->\n",
    "\n",
    "<!-- 9 Inductive Data Types -->\n",
    "\n",
    "<!-- 11 Concrete Syntax -->\n",
    "\n",
    "<!-- 12 Abstract Syntax and Parsing -->\n",
    "\n",
    "<!-- 13 Exercise: Syntax -->\n",
    "\n",
    "<!-- 14 Static Scoping -->\n",
    "\n",
    "<!-- 15 Judgments -->\n",
    "\n",
    "<!-- 16 Variables, Basic Values, and Judgments Lab -->\n",
    "\n",
    "<!-- 18 Operational Semantics -->\n",
    "\n",
    "<!-- 19 Functions and Dynamic Scoping -->\n",
    "\n",
    "<!-- 20 Big-Step Exercise -->\n",
    "\n",
    "<!-- 21 Evaluation Order  -->\n",
    "\n",
    "### Learning Goals\n",
    "\n",
    "The primary learning goals of this assignment are to build intuition for\n",
    "the following:\n",
    "\n",
    "-   the distinction between a big-step and a small-step operational\n",
    "    semantics;\n",
    "-   evaluation order; and\n",
    "-   substitution and program transformation.\n",
    "\n",
    "Functional Programming Skills  \n",
    "Iteration. Introduction to higher-order functions.\n",
    "\n",
    "Programming Language Ideas  \n",
    "Semantics: evaluation order. Small-step operational semantics.\n",
    "Substitution and program transformation.\n",
    "\n",
    "### Instructions\n",
    "\n",
    "A version of project files for this lab resides in the public\n",
    "[pppl-lab3](https://github.com/csci3155/pppl-lab3) repository. Please\n",
    "follow separate instructions to get a private clone of this repository\n",
    "for your work.\n",
    "\n",
    "You will be replacing `???` or `case _ => ???` in the `Lab3.scala` file\n",
    "with solutions to the coding exercises described below.\n",
    "\n",
    "**Your lab will not be graded if it does not compile**. You may check\n",
    "compilation with your IDE, `sbt compile`, or with the “sbt compile”\n",
    "GitHub Action provided for you. Comment out any code that does not\n",
    "compile or causes a failing assert. Put in `???` as needed to get\n",
    "something that compiles without error.\n",
    "\n",
    "You may add additional tests to the `Lab3Spec.scala` file. In the\n",
    "`Lab3Spec.scala`, there is empty test class `Lab3StudentSpec` that you\n",
    "can use to separate your tests from the given tests in the `Lab3Spec`\n",
    "class. You are also likely to edit `Lab3.worksheet.sc` for any scratch\n",
    "work. You can also use `Lab3.worksheet.js` to write and experiment in a\n",
    "JavaScript file that you can then parse into a JavaScripty AST (see\n",
    "`Lab3.worksheet.sc`).\n",
    "\n",
    "If you like, you may use this notebook for experimentation. However,\n",
    "**please make sure your code is in `Lab3.scala`; code in this notebook\n",
    "will not graded.**\n",
    "\n",
    "Note that there is a section with concept exercises\n",
    "(<a href=\"#sec-concept-exercises-small-step-lab\"\n",
    "class=\"quarto-xref\">Section 6</a>). Make sure to complete the concept\n",
    "exercises in that section and **turn in this file as part of your\n",
    "submission for the concept exercises**. However, all code and testing\n",
    "exercises from other sections are submitted in `Lab3.scala` or\n",
    "`Lab3Spec.scala`.\n",
    "\n",
    "Recall that you need to switch kernels between running JavaScript and\n",
    "Scala cells.\n",
    "\n",
    "## Small-Step Interpreter: JavaScripty Functions\n",
    "\n",
    "We consider the same JavaScripty variant as in the previous exercise on\n",
    "big-step operational semantics (**?@sec-big-step-interpreter-exercise**)\n",
    "where the interesting language feature are first-class functions:"
   ]
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    "trait Expr\n",
    "case class Fun(xopt: Option[String], y: String, e1: Expr) extends Expr // e ::= x?(y) => e1\n",
    "case class Call(e1: Expr, e2: Expr) extends Expr                       // e ::= e1(e2) "
   ]
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    "We consider a `Fun` constructor for representing JavaScripty function\n",
    "literals. This version of `Fun` allows for named functions. When a\n",
    "function expression\n",
    "$x\\texttt{(}y\\texttt{)} \\mathrel{\\texttt{=}\\!\\texttt{>}} e'$ has a name,\n",
    "then it is can be recursive. As noted previously about recursive\n",
    "functions (**?@sec-recursive-functions**), variable $x$ is an additional\n",
    "formal parameter, and the function body $e'$ may have free variable uses\n",
    "of $x$. The variable $x$ gets bound to itself (i.e., the function value\n",
    "for $x\\texttt{(}y\\texttt{)} \\mathrel{\\texttt{=}\\!\\texttt{>}} e'$) on a\n",
    "function call.\n",
    "\n",
    "In the abstract syntax representation, the `xopt: Option[String]`\n",
    "parameter in our `Fun` constructor is `None` if there is no identifier\n",
    "present (cannot be used recursively), or `Some(x: String)` if there is\n",
    "an identifier, `x`, present (can be used recursively).\n",
    "\n",
    "In this lab, we will do two things. First, we will move to implementing\n",
    "a small-step interpreter with a function `step` that takes an `e: Expr`\n",
    "and returns a one-step reduction of `e`. A small-step interpreter makes\n",
    "explicit the evaluation order of expressions. Second, we will remove\n",
    "environments and instead use a language semantics based on substitution.\n",
    "This change will result in static, lexical scoping without needing\n",
    "closures, thus demonstrating another way to fix dynamical scoping.\n",
    "\n",
    "These two changes are orthogonal, that is, one could implement a\n",
    "big-step interpreter using substitution (as in\n",
    "**?@sec-big-step-substitution**) or a small-step interpreter using\n",
    "environments. Substitution is a fairly simple way to get lexical\n",
    "scoping, but in practice, it is rarely used because it is not the most\n",
    "efficient implementation.\n",
    "\n",
    "## Static Scoping\n",
    "\n",
    "<span class=\"theorem-title\">**Exercise 1 (Substitute)**</span> Since our\n",
    "implementation requires substitution, we begin by implementing\n",
    "`substitute`, which substitutes value `v` for all *free* occurrences of\n",
    "variable `x` in expression `e`:"
   ]
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   "source": [
    "def substitute(e: Expr, v: Expr, x: String): Expr = ???"
   ]
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    "We advise defining `substitute` by induction on `e`. The cases to be\n",
    "careful about are `ConstDecl` and `Fun` because these are the variable\n",
    "binding constructs (as discussed in the reading on substitution in\n",
    "**?@sec-substitution**). In particular, calling `substitute` on\n",
    "expression\n",
    "\n",
    "``` typescript\n",
    "a; { const a = 4; a }\n",
    "```\n",
    "\n",
    "with value `3` for variable `a` should return\n",
    "\n",
    "``` typescript\n",
    "3; { const a = 4; a }\n",
    "```\n",
    "\n",
    "not\n",
    "\n",
    "``` typescript\n",
    "3; { const a = 4; 3 }\n",
    "```\n",
    "\n",
    "This function is a helper for the `step` function, but you might want to\n",
    "implement all of the cases of `step` that do not require `substitute`\n",
    "first.\n",
    "\n",
    "## Iteration\n",
    "\n",
    "Our `step` performs a single reduction step. We may want to test it by\n",
    "repeatedly calling it with an expression until reducing to value. Thus,\n",
    "from a software engineering standpoint, you might want to evolve the\n",
    "`iterate` function described below together with your implementation of\n",
    "`step`.\n",
    "\n",
    "This idea of repeatedly performing an action until some condition is\n",
    "satisfied is a loop or iteration. We have seen that we can iterate with\n",
    "a tail-recursive helper function. For example, consider the `sumTo`\n",
    "function that sums the integers from `0` to `n`:"
   ]
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    "def sumTo(n: Int): Int = {\n",
    "  def loop(acc: Int, i: Int): Int = {\n",
    "    require(n >= 0)\n",
    "    if (i > n) acc\n",
    "    else loop(acc + i, i + 1)\n",
    "  }\n",
    "  loop(0, 0)\n",
    "}\n",
    "sumTo(100)"
   ]
  },
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    "This pattern of repeating something until a condition is satisfied is\n",
    "exceedingly common (e.g., computing the square root using Newton-Raphson\n",
    "approximation until the error is small enough from a previous\n",
    "assignment).\n",
    "\n",
    "Because this pattern is so common, we want to get practice refactoring\n",
    "this pattern into a library function. This library function will be a\n",
    "higher-order function because it takes the “something” (i.e., what to do\n",
    "in each loop iteration) as a function parameter.\n",
    "\n",
    "<span class=\"theorem-title\">**Exercise 2 (Iterate with Error\n",
    "Side-Effects)**</span> Implement the generic, higher-order library\n",
    "function `iterateBasic`. The `iterateBasic` function repeatedly calls\n",
    "(i.e., iterates) the callback `stepi` until it returns `None` starting\n",
    "from `acc0: A`. Note that `iterateBasic` is generic in the accumulation\n",
    "type `A`. The `stepi` callback takes the current accumulator of type `A`\n",
    "and the iteration number as an `Int` and indicates continuing by\n",
    "returning `Some(acc)` for some next accumulator value `acc`."
   ]
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   "source": [
    "def iterateBasic[A](acc0: A)(stepi: (A, Int) => Option[A]): A = {\n",
    "  def loop(acc: A, i: Int): A = ???\n",
    "  loop(acc0, 0)\n",
    "}"
   ]
  },
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   "source": [
    "We can test `iterateBasic` by using it with a client like `sumTo`:"
   ]
  },
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    "def sumTo(n: Int) = {\n",
    "  iterateBasic(0) { case (acc, i) =>\n",
    "    require(n >= 0)\n",
    "    if (i > n) None\n",
    "    else Some(acc + i)\n",
    "  }\n",
    "}\n",
    "sumTo(100)"
   ]
  },
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   "id": "db8e7531-05cf-41d9-9313-5787dfe7d75a",
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    "We see how `sumTo` can use `iterateBasic`.\n",
    "\n",
    "<span class=\"theorem-title\">**Exercise 3 (Iterate with Error\n",
    "Values)**</span> One unfortunate aspect of the above is that `sumTo`\n",
    "“exits `iterateBasic` with an error” by throwing an exception (i.e.,\n",
    "with the `require(n >= 0)`). Let us refactor `iterateBasic` to allow for\n",
    "explicit error values using `Either[Err, A]`:"
   ]
  },
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   "id": "40f533dc",
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   "source": [
    "def iterate[Err, A](acc0: A)(stepi: (A, Int) => Option[Either[Err, A]]): Either[Err, A] = {\n",
    "  def loop(acc: A, i: Int): Either[Err, A] = ???\n",
    "  loop(acc0, 0)\n",
    "}\n",
    "\n",
    "def sumTo(n: Int): Either[IllegalArgumentException, Int] = {\n",
    "  iterate(0) { case (acc, i) =>\n",
    "    if (n < 0) Some(Left(new IllegalArgumentException(\"requirement failed\")))\n",
    "    else if (i > n) None\n",
    "    else Some(Right(acc + i))\n",
    "  }\n",
    "}\n",
    "sumTo(100)\n",
    "sumTo(-1)"
   ]
  },
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   "cell_type": "markdown",
   "id": "e79c6d11-62b5-463b-9d50-dc824ec0ade7",
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    "The `iterate` is now parametrized by an error type `Err` and returns an\n",
    "`Either[Err, A]`. The `stepi` callback should return `None` if it wants\n",
    "to stop normally, `Some(Left(err))` if it wants to stop with an error,\n",
    "and `Some(Right(acc)` if it wants to continue with an accumulator value\n",
    "`acc`.\n",
    "\n",
    "We can now see how we can use `iterate` as a library function to iterate\n",
    "your `step` implementation. In particular, this is how `iterate` will be\n",
    "used to iterate `step` while adding some debugging output:\n",
    "\n",
    "``` scala\n",
    "def iterateStep(e: Expr) = {\n",
    "  require(closed(e), s\"iterateStep: ${e} not closed\")\n",
    "  if (debug) {\n",
    "    println(\"------------------------------------------------------------\")\n",
    "    println(\"Evaluating with step ...\")\n",
    "  }\n",
    "  val v = iterate(e) { (e: Expr, n: Int) =>\n",
    "    if (debug) { println(s\"Step $n: $e\") }\n",
    "    if (isValue(e)) None else Some(step(e))\n",
    "  }\n",
    "  if (debug) { println(\"Value: \" + v) }\n",
    "  v\n",
    "}\n",
    "```\n",
    "\n",
    "Of particular interest is the anonymous function passed to `iterate`\n",
    "that calls your implementation of `step`.\n",
    "\n",
    "## Small-Step Interpreter\n",
    "\n",
    "In this section, we implement the one-step evaluation judgment form\n",
    "$e \\longrightarrow r$ that says, “Expression $e$ can take one step of\n",
    "evaluation to a step-result $r$.”\n",
    "\n",
    "$$\n",
    "\\begin{array}{rrrl}\n",
    "\\text{step-results} & r& \\mathrel{::=}& \\mathop{\\mathsf{typeerror}}e \\mid e'\n",
    "\\end{array}\n",
    "$$\n",
    "\n",
    "A step-result $r$ is either a $\\mathop{\\mathsf{typeerror}}e$ indicating\n",
    "a dynamic type error in attempting to reduce $e$ or a successful\n",
    "one-step reduction to an expression $e'$.\n",
    "\n",
    "We represent a step-result $r$ in Scala using a type\n",
    "`Either[DynamicTypeError, Expr]`:"
   ]
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   "source": [
    "case class DynamicTypeError(e: Expr) {\n",
    "  override def toString = s\"TypeError: in expression $e\"\n",
    "}\n",
    "type Result = Either[DynamicTypeError, Expr] // r ::= typeerror e | e"
   ]
  },
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    "Note that unlike before, `DynamicTypeError` is not an `Exception`, so it\n",
    "cannot be `throw`n.\n",
    "\n",
    "The small-step semantics that we should implement are given in the\n",
    "section below (<a href=\"#sec-small-step-operational-semantics\"\n",
    "class=\"quarto-xref\">Section 5</a>). The language we implement is\n",
    "JavaScripty with numbers, booleans, strings, $\\mathbf{undefined}$,\n",
    "printing, and first-class functions. It is a simpler language than the\n",
    "previous lab because we remove type coercions (except to booleans) and\n",
    "replace most coercion cases with dynamic type error\n",
    "$\\mathop{\\mathsf{typeerror}}e$.\n",
    "\n",
    "<span class=\"theorem-title\">**Exercise 4 (Step without Dynamic Type\n",
    "Checking)**</span> We advise first implementing the cases restricted to\n",
    "judgments of the form $e \\longrightarrow e'$, that is, implement the\n",
    "$\\TirName{Do}$ and $\\TirName{Search}$ rules while ignoring the\n",
    "$\\TirName{TypeError}$ and $\\TirName{Propagate}$ rules. Start with\n",
    "implementing a `stepBasic` function with type:"
   ]
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   "source": [
    "def stepBasic(e: Expr): Expr = ???"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "That is, just crash with a `MatchError` exception if your `step`\n",
    "encounters any ill-typed expression `e`.\n",
    "\n",
    "The suggested practice here is to read some rules, write a few tests for\n",
    "those rules, and implement the cases for those tests.\n",
    "\n",
    "<span class=\"theorem-title\">**Exercise 5 (Step with Dynamic Type\n",
    "Checking)**</span> Then, copy your code from `stepBasic` to `stepCheck`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "82d9afec",
   "metadata": {},
   "outputs": [],
   "source": [
    "def stepCheck(e: Expr): Either[DynamicTypeError, Expr] = ???"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3540d43d-6e7a-495f-b476-96f4bba30c17",
   "metadata": {},
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    "to then add dynamic type checking. You will likely need to refactor your\n",
    "code to satisfy the new types before implementing the\n",
    "$\\TirName{TypeError}$ and $\\TirName{Propagate}$ rules.\n",
    "\n",
    "<span class=\"theorem-title\">**Exercise 6 (To Boolean)**</span> You will\n",
    "need to implement a `toBoolean` function to convert JavaScripty values\n",
    "to booleans, following the $\\TirName{ToBoolean}$ rules in\n",
    "<a href=\"#sec-small-step-operational-semantics\"\n",
    "class=\"quarto-xref\">Section 5</a>."
   ]
  },
  {
   "cell_type": "code",
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   "source": [
    "def toBoolean(e: Expr): Boolean = ???"
   ]
  },
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    "However, you will not need any other type coercion functions here.\n",
    "\n",
    "#### Notes\n",
    "\n",
    "-   Note that the tests call the `step` function that is originally\n",
    "    defined as:\n",
    "\n",
    "    ``` scala\n",
    "    //def step(e: Expr): Either[DynamicTypeError, Expr] = Right(stepBasic(e))\n",
    "    def step(e: Expr): Either[DynamicTypeError, Expr] = stepCheck(e)\n",
    "    ```\n",
    "\n",
    "    You can first test `stepBasic` by uncommenting the first line and\n",
    "    commenting out the second line.\n",
    "\n",
    "-   Note that the provided tests are minimal. You will want to add your\n",
    "    own tests to cover most language features.\n",
    "\n",
    "## Small-Step Operational Semantics\n",
    "\n",
    "In this section, we give the small-step operational semantics for\n",
    "JavaScripty with numbers, booleans, strings, $\\mathbf{undefined}$,\n",
    "printing, and first-class functions. We have type coercions to booleans\n",
    "but otherwise use dynamic type error for other cases.\n",
    "\n",
    "We write ${}[v/x]e$ for substituting value $v$ for all free occurrences\n",
    "of the variable $x$ in expression $e$ (i.e., a call to `substitute`).\n",
    "\n",
    "It is informative to compare the small-step semantics used in this lab\n",
    "and the big-step semantics from last homework.\n",
    "\n",
    "### Do Rules\n",
    "\n",
    "$\\fbox{$e \\longrightarrow e'$}$\n",
    "\n",
    "$\\inferrule[DoNeg]{\n",
    "  n' = - n_1\n",
    "}{\n",
    "  \\mathop{\\texttt{-}} n_1 \\longrightarrow n'\n",
    "}$\n",
    "\n",
    "$\\inferrule[DoArith]{\n",
    "  n' = n_1 \\mathbin{\\mathit{bop}}n_2\n",
    "  \\and\n",
    "  \\mathit{bop}\\in \\left\\{  \\texttt{+}, \\texttt{-}, \\texttt{*}, \\texttt{/} \\right\\}\n",
    "}{\n",
    "  n_1 \\mathbin{\\mathit{bop}}n_2 \\longrightarrow n'\n",
    "}$\n",
    "\n",
    "$\\inferrule[DoPlusString]{\n",
    "  \\mathit{str}' = \\mathit{str}_1 \\mathit{str}_2\n",
    "}{\n",
    "  \\mathit{str}_1 \\mathbin{\\texttt{+}} \\mathit{str}_2 \\longrightarrow\\mathit{str}'\n",
    "}$\n",
    "\n",
    "$\\inferrule[DoInequalityNumber]{\n",
    "  b' = n_1 \\mathbin{\\mathit{bop}}n_2\n",
    "  \\and\n",
    "  \\mathit{bop}\\in \\left\\{  \\texttt{<}, \\texttt{<=}, \\texttt{>}, \\texttt{>=} \\right\\}\n",
    "}{\n",
    "  n_1 \\mathbin{\\mathit{bop}}n_2 \\longrightarrow b'\n",
    "}$\n",
    "\n",
    "$\\inferrule[DoInequalityString]{\n",
    "  b' = \\mathit{str}_1 \\mathbin{\\mathit{bop}}\\mathit{str}_2\n",
    "  \\and\n",
    "  \\mathit{bop}\\in \\left\\{  \\texttt{<}, \\texttt{<=}, \\texttt{>}, \\texttt{>=} \\right\\}\n",
    "}{\n",
    "  \\mathit{str}_1 \\mathbin{\\mathit{bop}}\\mathit{str}_2 \\longrightarrow b'\n",
    "}$\n",
    "\n",
    "$\\inferrule[DoEquality]{\n",
    "  b' = (v_1 \\mathbin{\\mathit{bop}}v_2)\n",
    "  \\and\n",
    "  \\mathit{bop}\\in \\left\\{  \\texttt{===}, \\texttt{!==} \\right\\}\n",
    "}{\n",
    "  v_1 \\mathbin{\\mathit{bop}}v_2 \\longrightarrow b'\n",
    "}$\n",
    "\n",
    "$\\inferrule[DoNot]{\n",
    "  v_1 \\rightsquigarrow b_1\n",
    "}{\n",
    "  \\mathop{\\texttt{!}} v_1 \\longrightarrow\\neg b_1\n",
    "}$\n",
    "\n",
    "$\\inferrule[DoAndTrue]{\n",
    "  v_1 \\rightsquigarrow \\mathbf{true}\n",
    "}{\n",
    "  v_1 \\mathbin{\\texttt{\\&\\&}} e_2 \\longrightarrow e_2\n",
    "}$\n",
    "\n",
    "$\\inferrule[DoAndFalse]{\n",
    "  v_1 \\rightsquigarrow \\mathbf{false}\n",
    "}{\n",
    "  v_1 \\mathbin{\\texttt{\\&\\&}} e_2 \\longrightarrow v_1\n",
    "}$\n",
    "\n",
    "$\\inferrule[DoOrTrue]{\n",
    "  v_1 \\rightsquigarrow \\mathbf{true}\n",
    "}{\n",
    "  v_1 \\mathbin{\\texttt{||}} e_2 \\longrightarrow v_1\n",
    "}$\n",
    "\n",
    "$\\inferrule[DoOrFalse]{\n",
    "  v_1 \\rightsquigarrow \\mathbf{false}\n",
    "}{\n",
    "  v_1 \\mathbin{\\texttt{||}} e_2 \\longrightarrow e_2\n",
    "}$\n",
    "\n",
    "$\\inferrule[DoIfTrue]{\n",
    "  v_1 \\rightsquigarrow \\mathbf{true}\n",
    "}{\n",
    "  v_1\\;\\texttt{?}\\;e_2\\;\\texttt{:}\\;e_3 \\longrightarrow e_2\n",
    "}$\n",
    "\n",
    "$\\inferrule[DoIfFalse]{\n",
    "  v_1 \\rightsquigarrow \\mathbf{false}\n",
    "}{\n",
    "  v_1\\;\\texttt{?}\\;e_2\\;\\texttt{:}\\;e_3 \\longrightarrow e_3\n",
    "}$\n",
    "\n",
    "$\\inferrule[DoSeq]{\n",
    "}{\n",
    "  v_1 \\mathbin{\\texttt{,}} e_2 \\longrightarrow e_2\n",
    "}$\n",
    "\n",
    "$\\inferrule[DoPrint]{\n",
    "  \\text{$v_1$ printed}\n",
    "}{\n",
    "  \\texttt{console}\\texttt{.}\\texttt{log}\\texttt{(}v_1\\texttt{)} \\longrightarrow\\mathbf{undefined}\n",
    "}$\n",
    "\n",
    "$\\inferrule[DoConst]{\n",
    "}{\n",
    "  \\mathbf{const}\\;x\\;\\texttt{=}\\;v_1\\texttt{;}\\;e_2 \\longrightarrow[v_1/x]e_2\n",
    "}$\n",
    "\n",
    "$\\inferrule[DoCall]{\n",
    "}{\n",
    "   (\\texttt{(}x\\texttt{)} \\mathrel{\\texttt{=}\\!\\texttt{>}} e_1) \\texttt{(}v_2\\texttt{)} \\longrightarrow{}[v_2/x]e_1\n",
    "}$\n",
    "\n",
    "$\\inferrule[DoCallRec]{\n",
    "  v_1 = (x_1\\texttt{(}x_2\\texttt{)} \\mathrel{\\texttt{=}\\!\\texttt{>}} e_1)\n",
    "}{\n",
    "   v_1\\texttt{(}v_2\\texttt{)}  \\longrightarrow {}[v_1/x_1] {}[v_2/x_2]e_1  \n",
    "}$\n",
    "\n",
    "Figure 1: The $\\TirName{Do}$ rules for JavaScripty with numbers,\n",
    "booleans, strings, $\\mathbf{undefined}$, printing, and first-class\n",
    "functions.\n",
    "\n",
    "### Search Rules\n",
    "\n",
    "$\\fbox{$e \\longrightarrow e'$}$\n",
    "\n",
    "$\\inferrule[SearchUnary]{\n",
    "  e_1 \\longrightarrow e_1'\n",
    "}{\n",
    "  \\mathop{\\mathit{uop}}e_1 \\longrightarrow\\mathop{\\mathit{uop}}e_1'\n",
    "}$\n",
    "\n",
    "$\\inferrule[SearchBinary1]{\n",
    "  e_1 \\longrightarrow e_1'\n",
    "}{\n",
    "  e_1 \\mathbin{\\mathit{bop}}e_2 \\longrightarrow e_1' \\mathbin{\\mathit{bop}}e_2\n",
    "}$\n",
    "\n",
    "$\\inferrule[SearchBinary2]{\n",
    "  e_2 \\longrightarrow e_2'\n",
    "}{\n",
    "  v_1 \\mathbin{\\mathit{bop}}e_2 \\longrightarrow v_1 \\mathbin{\\mathit{bop}}e_2'\n",
    "}$\n",
    "\n",
    "$\\inferrule[SearchIf]{\n",
    "  e_1 \\longrightarrow e_1'\n",
    "}{\n",
    "  e_1\\;\\texttt{?}\\;e_2\\;\\texttt{:}\\;e_3 \\longrightarrow e_1'\\;\\texttt{?}\\;e_2\\;\\texttt{:}\\;e_3\n",
    "}$\n",
    "\n",
    "$\\inferrule[SearchPrint]{\n",
    "  e_1 \\longrightarrow e_1'\n",
    "}{\n",
    "  \\texttt{console}\\texttt{.}\\texttt{log}\\texttt{(}e_1\\texttt{)} \\longrightarrow\\texttt{console}\\texttt{.}\\texttt{log}\\texttt{(}e_1'\\texttt{)}\n",
    "}$\n",
    "\n",
    "$\\inferrule[SearchConst]{\n",
    "  e_1 \\longrightarrow e_1'\n",
    "}{\n",
    "  \\mathbf{const}\\;x\\;\\texttt{=}\\;e_1\\texttt{;}\\;e_2 \\longrightarrow\\mathbf{const}\\;x\\;\\texttt{=}\\;e_1'\\texttt{;}\\;e_2  \n",
    "}$\n",
    "\n",
    "$\\inferrule[SearchCall1]{\n",
    "  e_1 \\longrightarrow e_1'\n",
    "}{\n",
    "  e_1\\texttt{(}e_2\\texttt{)} \\longrightarrow e_1'\\texttt{(}e_2\\texttt{)}\n",
    "}$\n",
    "\n",
    "$\\inferrule[SearchCall2]{\n",
    "  e_2 \\longrightarrow e_2'\n",
    "}{\n",
    "   v_1 \\texttt{(}e_2\\texttt{)} \\longrightarrow v_1 \\texttt{(}e_2'\\texttt{)}\n",
    "}$\n",
    "\n",
    "Figure 2: The $\\TirName{Search}$ rules for JavaScripty with numbers,\n",
    "booleans, strings, $\\mathbf{undefined}$, printing and first-class\n",
    "functions.\n",
    "\n",
    "### Coercing to Boolean\n",
    "\n",
    "$\\fbox{$v \\rightsquigarrow b$}$\n",
    "\n",
    "$\\inferrule[ToBooleanNumFalse]{\n",
    "  n= 0.0 \\lor n= -0.0 \\lor n= \\mathsf{NaN}\n",
    "}{\n",
    "  n \\rightsquigarrow \\mathbf{false}\n",
    "}$\n",
    "\n",
    "$\\inferrule[ToBooleanNumTrue]{\n",
    "  n\\neq 0.0 \\and n\\neq -0.0 \\and n\\neq \\mathsf{NaN}\n",
    "}{\n",
    "  n \\rightsquigarrow \\mathbf{true}\n",
    "}$\n",
    "\n",
    "$\\inferrule[ToBooleanBoolean]{\n",
    "}{\n",
    "  b \\rightsquigarrow b\n",
    "}$\n",
    "\n",
    "$\\inferrule[ToBooleanStrFalse]{\n",
    "}{\n",
    "   \\texttt{\"\"}  \\rightsquigarrow \\mathbf{false}\n",
    "}$\n",
    "\n",
    "$\\inferrule[ToBooleanStrTrue]{\n",
    "  \\mathit{str}\\neq \\texttt{\"\"}\n",
    "}{\n",
    "  \\mathit{str} \\rightsquigarrow \\mathbf{true}\n",
    "}$\n",
    "\n",
    "$\\inferrule[ToBooleanUndefined]{\n",
    "}{\n",
    "  \\mathbf{undefined} \\rightsquigarrow \\mathbf{false}\n",
    "}$\n",
    "\n",
    "$\\inferrule[ToBooleanFun]{\n",
    "}{\n",
    "  x^{?}\\texttt{(}y\\texttt{)} \\mathrel{\\texttt{=}\\!\\texttt{>}} e \\rightsquigarrow \\mathbf{true}\n",
    "}$\n",
    "\n",
    "Figure 3: The $\\TirName{ToBoolean}$ rules for JavaScripty with numbers,\n",
    "booleans, strings, $\\mathbf{undefined}$, and first-class functions.\n",
    "\n",
    "### Dynamic Typing Rules\n",
    "\n",
    "$\\fbox{$e \\longrightarrow r$}$\n",
    "\n",
    "$\\inferrule[TypeErrorNeg]{\n",
    "  v_1 \\neq n_1\n",
    "}{\n",
    "   \\mathop{\\texttt{-}} v_1  \\longrightarrow \\mathop{\\mathsf{typeerror}}(\\mathop{\\texttt{-}} v_1) \n",
    "}$\n",
    "\n",
    "$\\inferrule[TypeErrorArith2]{\n",
    "  v_2 \\neq n_2\n",
    "  \\and \n",
    "  \\mathit{bop}\\in \\left\\{  \\texttt{+}, \\texttt{-}, \\texttt{*}, \\texttt{/} \\right\\}\n",
    "}{\n",
    "   n_1 \\mathbin{\\mathit{bop}}v_2  \\longrightarrow \\mathop{\\mathsf{typeerror}}(n_1 \\mathbin{\\mathit{bop}}v_2) \n",
    "}$\n",
    "\n",
    "$\\inferrule[TypeErrorArith1]{\n",
    "  v_1 \\neq n_1\n",
    "  \\and\n",
    "  \\mathit{bop}\\in \\left\\{  \\texttt{+}, \\texttt{-}, \\texttt{*}, \\texttt{/} \\right\\}\n",
    "}{\n",
    "   v_1 \\mathbin{\\mathit{bop}}e_2  \\longrightarrow \\mathop{\\mathsf{typeerror}}(v_1 \\mathbin{\\mathit{bop}}e_2) \n",
    "}$\n",
    "\n",
    "$\\inferrule[TypeErrorPlusString2]{\n",
    "  v_2 \\neq \\mathit{str}_2\n",
    "}{\n",
    "   \\mathit{str}_1 \\mathbin{\\texttt{+}} v_2  \\longrightarrow \\mathop{\\mathsf{typeerror}}(\\mathit{str}_1 \\mathbin{\\texttt{+}} v_2) \n",
    "}$\n",
    "\n",
    "$\\inferrule[TypeErrorPlusString1]{\n",
    "  v_1 \\neq \\mathit{str}_1\n",
    "}{\n",
    "   v_1 \\mathbin{\\texttt{+}} e_2  \\longrightarrow \\mathop{\\mathsf{typeerror}}(v_1 \\mathbin{\\texttt{+}} e_2) \n",
    "}$\n",
    "\n",
    "$\\inferrule[TypeErrorInequalityNumber2]{\n",
    "  v_2 \\neq n_2\n",
    "  \\and \n",
    "  \\mathit{bop}\\in \\left\\{  \\texttt{<}, \\texttt{<=}, \\texttt{>}, \\texttt{>=} \\right\\}\n",
    "}{\n",
    "  n_1 \\mathbin{\\mathit{bop}}v_2  \\longrightarrow \\mathop{\\mathsf{typeerror}}(n_1 \\mathbin{\\mathit{bop}}v_2) \n",
    "}$\n",
    "\n",
    "$\\inferrule[TypeErrorInequalityNumber1]{\n",
    "  v_1 \\neq n_1\n",
    "  \\and\n",
    "  \\mathit{bop}\\in \\left\\{  \\texttt{<}, \\texttt{<=}, \\texttt{>}, \\texttt{>=} \\right\\}\n",
    "}{\n",
    "   v_1 \\mathbin{\\mathit{bop}}e_2  \\longrightarrow \\mathop{\\mathsf{typeerror}}(v_1 \\mathbin{\\mathit{bop}}e_2) \n",
    "}$\n",
    "\n",
    "$\\inferrule[TypeErrorInequalityString2]{\n",
    "  v_2 \\neq \\mathit{str}_2\n",
    "  \\and\n",
    "  \\mathit{bop}\\in \\left\\{  \\texttt{<}, \\texttt{<=}, \\texttt{>}, \\texttt{>=} \\right\\}\n",
    "}{\n",
    "   \\mathit{str}_1 \\mathbin{\\mathit{bop}}v_2  \\longrightarrow \\mathop{\\mathsf{typeerror}}(\\mathit{str}_1 \\mathbin{\\mathit{bop}}v_2 ) \n",
    "}$\n",
    "\n",
    "$\\inferrule[TypeErrorInequalityString1]{\n",
    "  v_1 \\neq \\mathit{str}_1\n",
    "  \\and\n",
    "  \\mathit{bop}\\in \\left\\{  \\texttt{<}, \\texttt{<=}, \\texttt{>}, \\texttt{>=} \\right\\}\n",
    "}{\n",
    "   v_1 \\mathbin{\\mathit{bop}}e_2  \\longrightarrow \\mathop{\\mathsf{typeerror}}(v_1 \\mathbin{\\mathit{bop}}e_2 ) \n",
    "}$\n",
    "\n",
    "$\\inferrule[TypeErrorCall]{\n",
    "  v_1 \\neq x^{?}\\texttt{(}y\\texttt{)} \\mathrel{\\texttt{=}\\!\\texttt{>}} e_1\n",
    "}{\n",
    "  v_1\\texttt{(}e_2\\texttt{)} \\longrightarrow \\mathop{\\mathsf{typeerror}} (v_1\\texttt{(}e_2\\texttt{)})  \n",
    "}$\n",
    "\n",
    "$\\inferrule[PropagateUnary]{\n",
    "   e_1  \\longrightarrow \\mathop{\\mathsf{typeerror}}e \n",
    "}{\n",
    "   \\mathop{\\mathit{uop}} e_1  \\longrightarrow \\mathop{\\mathsf{typeerror}}e \n",
    "}$\n",
    "\n",
    "$\\inferrule[PropagateBinary1]{\n",
    "   e_1  \\longrightarrow \\mathop{\\mathsf{typeerror}}e \n",
    "}{\n",
    "   e_1 \\mathbin{\\mathit{bop}} e_2  \\longrightarrow \\mathop{\\mathsf{typeerror}}e \n",
    "}$\n",
    "\n",
    "$\\inferrule[PropagateBinary2]{\n",
    "   e_2  \\longrightarrow \\mathop{\\mathsf{typeerror}}e \n",
    "}{\n",
    "   v_1 \\mathbin{\\mathit{bop}} e_2  \\longrightarrow \\mathop{\\mathsf{typeerror}}e \n",
    "}$\n",
    "\n",
    "$\\inferrule[PropagateIf]{\n",
    "  e_1 \\longrightarrow \\mathop{\\mathsf{typeerror}}e \n",
    "}{\n",
    "  e_1\\;\\texttt{?}\\;e_2\\;\\texttt{:}\\;e_3 \\longrightarrow \\mathop{\\mathsf{typeerror}}e \n",
    "}$\n",
    "\n",
    "$\\inferrule[PropagatePrint]{\n",
    "   e_1  \\longrightarrow \\mathop{\\mathsf{typeerror}}e \n",
    "}{\n",
    "   \\texttt{console}\\texttt{.}\\texttt{log}\\texttt{(}e_1\\texttt{)}  \\longrightarrow \\mathop{\\mathsf{typeerror}}e \n",
    "}$\n",
    "\n",
    "$\\inferrule[PropagateConst]{\n",
    "   e_1  \\longrightarrow \\mathop{\\mathsf{typeerror}}e \n",
    "}{\n",
    "   \\mathbf{const}\\;x\\;\\texttt{=}\\;e_1\\texttt{;}\\;e_2  \\longrightarrow \\mathop{\\mathsf{typeerror}}e \n",
    "}$\n",
    "\n",
    "$\\inferrule[PropagateCall1]{\n",
    "  e_1 \\longrightarrow \\mathop{\\mathsf{typeerror}}e \n",
    "}{\n",
    "  e_1\\texttt{(}e_2\\texttt{)} \\longrightarrow \\mathop{\\mathsf{typeerror}}e \n",
    "}$\n",
    "\n",
    "$\\inferrule[PropagateCall2]{\n",
    "  e_2 \\longrightarrow \\mathop{\\mathsf{typeerror}}e \n",
    "}{\n",
    "   v_1\\texttt{(}e_2\\texttt{)}  \\longrightarrow \\mathop{\\mathsf{typeerror}}e \n",
    "}$\n",
    "\n",
    "Figure 4: The $\\TirName{TypeError}$ and $\\TirName{Propagate}$ rules for\n",
    "JavaScripty with numbers, booleans, strings, $\\mathbf{undefined}$,\n",
    "printing, and first-class functions.\n",
    "\n",
    "## Concept Exercises\n",
    "\n",
    "Make sure to complete the concept exercises in this section and turn in\n",
    "this file as part of your submission. However, all code and testing\n",
    "exercises from other sections are submitted in `Lab3.scala` or\n",
    "`Lab3Spec.scala`.\n",
    "\n",
    "<span class=\"theorem-title\">**Exercise 7 (Evaluation Order)**</span>\n",
    "Consider the small-step operational semantics shown in\n",
    "<a href=\"#sec-small-step-operational-semantics\"\n",
    "class=\"quarto-xref\">Section 5</a>. What is the evaluation order for\n",
    "$e_1 \\mathbin{\\texttt{+}} e_2$? Explain.\n",
    "\n",
    "**Edit this cell:**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a2bcc675-bf96-4ca2-968a-e4c7c9035ed1",
   "metadata": {},
   "source": [
    "???"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5f236a16-64ce-4ddd-ada8-c59c6f72be38",
   "metadata": {},
   "source": [
    "<span class=\"theorem-title\">**Exercise 8 (Changing Evaluation\n",
    "Order)**</span> How do we change the rules to obtain the opposite\n",
    "evaluation order?\n",
    "\n",
    "**Edit this cell:**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1ba8bb24-20da-433c-9e1f-2982484a3647",
   "metadata": {},
   "source": [
    "???"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3846d73e-5a0b-4cc2-adb7-5f4d4c9683ed",
   "metadata": {},
   "source": [
    "<span class=\"theorem-title\">**Exercise 9 (Using Short-Circuit\n",
    "Evaluation)**</span> Give an example that illustrates the usefulness of\n",
    "short-circuit evaluation. Explain your example.\n",
    "\n",
    "**Edit this cell:**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e09f1752-538e-48a7-954b-6b233d5602ea",
   "metadata": {},
   "source": [
    "???"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8b669537-c481-4050-a61a-fd0c896efc4c",
   "metadata": {},
   "source": [
    "<span class=\"theorem-title\">**Exercise 10 (Removing Short-Circuit\n",
    "Evaluation)**</span> Consider the small-step operational semantics shown\n",
    "in <a href=\"#sec-small-step-operational-semantics\"\n",
    "class=\"quarto-xref\">Section 5</a>. Does\n",
    "$e_1 \\mathbin{\\texttt{\\&\\&}} e_2$ short circuit? Explain. If\n",
    "$e_1 \\mathbin{\\texttt{\\&\\&}} e_2$ short circuits, give rules to\n",
    "eliminatesshort circuiting. If it does not short circuit, give the\n",
    "short-circuiting rules.\n",
    "\n",
    "**Edit this cell:**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "60a3d6cb-be4b-46ff-b2ec-28d5b44aa89a",
   "metadata": {},
   "source": [
    "???"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8cc99ffc-e92e-468e-9895-48472d9ae946",
   "metadata": {},
   "source": [
    "## Testing\n",
    "\n",
    "This section has some space to write some tests in our subset of\n",
    "JavaScript. You might want to work on these tests while you are\n",
    "implementing `step`. As before, you will add your tests to\n",
    "`Lab3StudentSpec`. Your interpreter will run the tests against the\n",
    "expected result you provide. We will write three tests, all of these\n",
    "tests must properly parse.\n",
    "\n",
    "<span class=\"theorem-title\">**Exercise 11 (Test 1: Higher-Order\n",
    "Function)**</span> Write a test case that has a function that takes a\n",
    "function value as an argument (i.e., is a higher order function):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "31950518-5470-4529-9cf1-6b3a5e6ebae8",
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "1555b8553ead2a857b18c9afd0b6db41",
     "grade": false,
     "grade_id": "test1-higher-order-function",
     "locked": false,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "???"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "70a1361a-5c18-4a3c-ab60-11c9470969af",
   "metadata": {},
   "source": [
    "<span class=\"theorem-title\">**Exercise 12 (Test 2: Recursion)**</span>\n",
    "Write a test case that uses recursion"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "89182178-318e-47bb-8d0c-9e82698cff1f",
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "914962b49a6beefb2f6d7313b0b52a87",
     "grade": false,
     "grade_id": "test2-recursion",
     "locked": false,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "???"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dcf17d0d-1ad3-4280-b50a-edebbf7e1fd0",
   "metadata": {},
   "source": [
    "<span class=\"theorem-title\">**Exercise 13 (Test 3: Any Test in this\n",
    "variant of JavaScripty)**</span> Write another test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "b8be895f-1af1-4106-958c-dd9763851eca",
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "3fca668afe5d69ccb211d30b0571ee11",
     "grade": false,
     "grade_id": "test3-anytest",
     "locked": false,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "???"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "032b8e70-75ff-4b08-aed4-6160955d55d2",
   "metadata": {},
   "source": [
    "#### Notes\n",
    "\n",
    "-   Remember to add these to `Lab3StudentSpec` in `Lab3Spec.scala`.\n",
    "-   Add the JavaScripty code as a string in `jsyStr` and the expected\n",
    "    result in `answer`.\n",
    "\n",
    "## Accelerated Component\n",
    "\n",
    "For the accelerated component of this lab, we will give rules and\n",
    "implement the behavior that enables us to match JavaScript semantics. In\n",
    "particular, we will give rules and implement type coercions for numbers\n",
    "and strings, and we will update our small-step operational semantics to\n",
    "use them.\n",
    "\n",
    "### Additional Type Coercions\n",
    "\n",
    "<span class=\"theorem-title\">**Exercise 14**</span> Give the inference\n",
    "rules defining the judgment form for coercing a value to a number\n",
    "$v \\rightsquigarrow n$\n",
    "\n",
    "**Edit this cell:**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6f3ee1ff-0a9e-46b4-8ce8-240ea775ab2c",
   "metadata": {},
   "source": [
    "???"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1f94c94c-4163-4978-aea9-14b4b8685c1f",
   "metadata": {},
   "source": [
    "<span class=\"theorem-title\">**Exercise 15**</span> Give the inference\n",
    "rules defining the judgment form for coercing a value to a string\n",
    "$v \\rightsquigarrow \\mathit{str}$\n",
    "\n",
    "**Edit this cell:**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9be104c8-a805-4cae-b078-639de87b1dd9",
   "metadata": {},
   "source": [
    "???"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "<span class=\"theorem-title\">**Exercise 16**</span> Implement the\n",
    "`toNumber` and `toStr`coercions."
   ]
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   "id": "5d05d408",
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    "def toNumber(v: Expr): Double = ???\n",
    "def toStr(v: Expr): String = ???"
   ]
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    "#### Notes\n",
    "\n",
    "-   If your recall Lab 2, we implemented these functions. They will be\n",
    "    the same here, except we must add the rules for functions.\n",
    "\n",
    "### Updating the Small-Step Operational Semantics\n",
    "\n",
    "Now that we are allowing type coersions, our operational semantics will\n",
    "change. For example, consider the following $\\TirName{Do}$ rule:\n",
    "\n",
    "$\\inferrule[DoNeg]{\n",
    "  v \\rightsquigarrow n\n",
    "}{\n",
    "  \\mathop{\\texttt{-}} v \\longrightarrow-n\n",
    "}$\n",
    "\n",
    "This is a new rule for $\\TirName{DoNeg}$, which is read as if $v$\n",
    "coerces to $n$, then $\\mathop{\\texttt{-}} v$ steps to $-n$. Now that we\n",
    "are allowing non-numbers to be coerced and then negated, we no longer\n",
    "have our $\\TirName{TypeErrorNeg}$ rule.\n",
    "\n",
    "Note that we only need to update $\\TirName{Do}$ rules with coercions and\n",
    "remove $\\TirName{TypeError}$ rules. We do not need to update the\n",
    "$\\TirName{Search}$ or $\\TirName{Propagage}$ rules.\n",
    "\n",
    "<span class=\"theorem-title\">**Exercise 17**</span> Explain why we only\n",
    "need to update the $\\TirName{Do}$ rules with coercions and remove\n",
    "$\\TirName{TypeError}$ rules, and we do not need to update the\n",
    "$\\TirName{Search}$ or $\\TirName{Propagage}$ rules.\n",
    "\n",
    "**Edit this cell:**"
   ]
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   "cell_type": "markdown",
   "id": "c20181aa-26c9-425a-ab32-800b3c55742d",
   "metadata": {},
   "source": [
    "???"
   ]
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   "id": "a377b034-1e94-483f-b369-1456ac217f6f",
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   "source": [
    "One rule of particular interest is $\\TirName{DoArith}$, which we need to\n",
    "split to account for $e_1 \\mathbin{\\texttt{+}} e_2$ being overloaded for\n",
    "numbers and strings. Given this, we need to rewrite $\\TirName{DoArith}$\n",
    "so it does not include $\\texttt{+}$ (and adds coersions), add the rule\n",
    "$\\TirName{DoPlusNumber}$, and alter $\\TirName{DoPlusString}$ to become\n",
    "two rules.\n",
    "\n",
    "<span class=\"theorem-title\">**Exercise 18**</span> Give these new\n",
    "$\\TirName{Do}$ rules\n",
    "\n",
    "***Edit this cell:***"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ba18b832-ce9c-4761-bbe2-7d394221be96",
   "metadata": {},
   "source": [
    "???"
   ]
  },
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   "cell_type": "markdown",
   "id": "82713e87-170b-44e1-a78f-c7c9bef197ea",
   "metadata": {},
   "source": [
    "Similar to this, our $\\TirName{DoInequalityNumber}$ rule must be split\n",
    "into two and altered:\n",
    "\n",
    "<span class=\"theorem-title\">**Exercise 19**</span> Give the two new\n",
    "$\\TirName{DoInequalityNumber\\(_1\\)}$ and\n",
    "$\\TirName{DoInequalityNumber\\(_2\\)}$ rules:\n",
    "\n",
    "***Edit this cell:***"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "27af0d9a-658e-4a58-bd33-b3bdd76d977f",
   "metadata": {},
   "source": [
    "???"
   ]
  },
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   "id": "e26a771b-ceb8-420b-b4e7-242cdd3128ab",
   "metadata": {},
   "source": [
    "Once we have all of these rules defined, we notice that most of our\n",
    "$\\mathsf{typeerror}$ rules no longer result in type errors. Therefore,\n",
    "most of them should be deleted. In fact, the only non-propagate rule\n",
    "left for type errors is $\\TirName{TypeErrorCall}$, since we are still\n",
    "not allowed to call something that is not a function (in JavaScript).\n",
    "\n",
    "### Update Step\n",
    "\n",
    "<span class=\"theorem-title\">**Exercise 20**</span> Now that we have our\n",
    "type conversion functions implemented and our new rules defined, we are\n",
    "ready to update `step`. Implement `stepCoerce` by first copying your\n",
    "code from `stepCheck` and then update based on your new rules."
   ]
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   "cell_type": "code",
   "execution_count": null,
   "id": "66926608",
   "metadata": {},
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   "source": [
    "def stepCoerce(e: Expr): Either[DynamicTypeError,Expr] = ???"
   ]
  },
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   "id": "a07344ab-334b-429e-a676-aaa344db8d26",
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   "source": [
    "As before, let the rules guide your implementation.\n",
    "\n",
    "### Notes\n",
    "\n",
    "-   None of your other functions should need to be altered.\n",
    "\n",
    "## Submission\n",
    "\n",
    "If you are a University of Colorado Boulder student, we use Gradescope\n",
    "for assignment submission. In summary,\n",
    "\n",
    "-   [ ] Create a private GitHub repository by clicking on a GitHub\n",
    "    Classroom link from the corresponding Canvas assignment entry.\n",
    "-   [ ] Clone your private GitHub repository to your development\n",
    "    environment (using the `<> Code` button on GitHub to get the\n",
    "    repository URL).\n",
    "-   [ ] Work on this lab from your cloned repository. Use Git to save\n",
    "    versions on GitHub (e.g., `git add`, `git commit`, `git push` on the\n",
    "    command line or via VSCode).\n",
    "-   [ ] Submit to the corresponding Gradescope assignment entry for\n",
    "    grading by choosing GitHub as the submission method.\n",
    "\n",
    "You need to have a GitHub identity and must have your full name in your\n",
    "GitHub profile in case we need to associate you with your submissions."
   ]
  }
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