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src/Doc/Prog_Prove/Types_and_funs.thy

author | nipkow |

Mon, 27 Jul 2020 15:58:43 +0200 | |

changeset 72307 | 1d6c3cba47fe |

parent 69609 | ff784d5a5bfb |

permissions | -rw-r--r-- |

unclear why I ever asked for type tree2

(*<*) theory Types_and_funs imports Main begin (*>*) text\<open> \vspace{-5ex} \section{Type and Function Definitions} Type synonyms are abbreviations for existing types, for example \index{string@\<open>string\<close>}\<close> type_synonym string = "char list" text\<open> Type synonyms are expanded after parsing and are not present in internal representation and output. They are mere conveniences for the reader. \subsection{Datatypes} \label{sec:datatypes} The general form of a datatype definition looks like this: \begin{quote} \begin{tabular}{@ {}rclcll} \indexed{\isacom{datatype}}{datatype} \<open>('a\<^sub>1,\<dots>,'a\<^sub>n)t\<close> & = & $C_1 \ \<open>"\<close>\tau_{1,1}\<open>"\<close> \dots \<open>"\<close>\tau_{1,n_1}\<open>"\<close>$ \\ & $|$ & \dots \\ & $|$ & $C_k \ \<open>"\<close>\tau_{k,1}\<open>"\<close> \dots \<open>"\<close>\tau_{k,n_k}\<open>"\<close>$ \end{tabular} \end{quote} It introduces the constructors \ $C_i :: \tau_{i,1}\Rightarrow \cdots \Rightarrow \tau_{i,n_i} \Rightarrow$~\<open>('a\<^sub>1,\<dots>,'a\<^sub>n)t\<close> \ and expresses that any value of this type is built from these constructors in a unique manner. Uniqueness is implied by the following properties of the constructors: \begin{itemize} \item \emph{Distinctness:} $C_i\ \ldots \neq C_j\ \dots$ \quad if $i \neq j$ \item \emph{Injectivity:} \begin{tabular}[t]{l} $(C_i \ x_1 \dots x_{n_i} = C_i \ y_1 \dots y_{n_i}) =$\\ $(x_1 = y_1 \land \dots \land x_{n_i} = y_{n_i})$ \end{tabular} \end{itemize} The fact that any value of the datatype is built from the constructors implies the \concept{structural induction}\index{induction} rule: to show $P~x$ for all $x$ of type \<open>('a\<^sub>1,\<dots>,'a\<^sub>n)t\<close>, one needs to show $P(C_i\ x_1 \dots x_{n_i})$ (for each $i$) assuming $P(x_j)$ for all $j$ where $\tau_{i,j} =$~\<open>('a\<^sub>1,\<dots>,'a\<^sub>n)t\<close>. Distinctness and injectivity are applied automatically by \<open>auto\<close> and other proof methods. Induction must be applied explicitly. Like in functional programming languages, datatype values can be taken apart with case expressions\index{case expression}\index{case expression@\<open>case ... of\<close>}, for example \begin{quote} \noquotes{@{term[source] "(case xs of [] \<Rightarrow> 0 | x # _ \<Rightarrow> Suc x)"}} \end{quote} Case expressions must be enclosed in parentheses. As an example of a datatype beyond \<^typ>\<open>nat\<close> and \<open>list\<close>, consider binary trees: \<close> datatype 'a tree = Tip | Node "'a tree" 'a "'a tree" text\<open>with a mirror function:\<close> fun mirror :: "'a tree \<Rightarrow> 'a tree" where "mirror Tip = Tip" | "mirror (Node l a r) = Node (mirror r) a (mirror l)" text\<open>The following lemma illustrates induction:\<close> lemma "mirror(mirror t) = t" apply(induction t) txt\<open>yields @{subgoals[display]} The induction step contains two induction hypotheses, one for each subtree. An application of \<open>auto\<close> finishes the proof. A very simple but also very useful datatype is the predefined @{datatype[display] option}\index{option@\<open>option\<close>}\index{None@\<^const>\<open>None\<close>}\index{Some@\<^const>\<open>Some\<close>} Its sole purpose is to add a new element \<^const>\<open>None\<close> to an existing type \<^typ>\<open>'a\<close>. To make sure that \<^const>\<open>None\<close> is distinct from all the elements of \<^typ>\<open>'a\<close>, you wrap them up in \<^const>\<open>Some\<close> and call the new type \<^typ>\<open>'a option\<close>. A typical application is a lookup function on a list of key-value pairs, often called an association list: \<close> (*<*) apply auto done (*>*) fun lookup :: "('a * 'b) list \<Rightarrow> 'a \<Rightarrow> 'b option" where "lookup [] x = None" | "lookup ((a,b) # ps) x = (if a = x then Some b else lookup ps x)" text\<open> Note that \<open>\<tau>\<^sub>1 * \<tau>\<^sub>2\<close> is the type of pairs, also written \<open>\<tau>\<^sub>1 \<times> \<tau>\<^sub>2\<close>. Pairs can be taken apart either by pattern matching (as above) or with the projection functions \<^const>\<open>fst\<close> and \<^const>\<open>snd\<close>: @{thm fst_conv[of x y]} and @{thm snd_conv[of x y]}. Tuples are simulated by pairs nested to the right: \<^term>\<open>(a,b,c)\<close> is short for \<open>(a, (b, c))\<close> and \<open>\<tau>\<^sub>1 \<times> \<tau>\<^sub>2 \<times> \<tau>\<^sub>3\<close> is short for \<open>\<tau>\<^sub>1 \<times> (\<tau>\<^sub>2 \<times> \<tau>\<^sub>3)\<close>. \subsection{Definitions} Non-recursive functions can be defined as in the following example: \index{definition@\isacom{definition}}\<close> definition sq :: "nat \<Rightarrow> nat" where "sq n = n * n" text\<open>Such definitions do not allow pattern matching but only \<open>f x\<^sub>1 \<dots> x\<^sub>n = t\<close>, where \<open>f\<close> does not occur in \<open>t\<close>. \subsection{Abbreviations} Abbreviations are similar to definitions: \index{abbreviation@\isacom{abbreviation}}\<close> abbreviation sq' :: "nat \<Rightarrow> nat" where "sq' n \<equiv> n * n" text\<open>The key difference is that \<^const>\<open>sq'\<close> is only syntactic sugar: after parsing, \<^term>\<open>sq' t\<close> is replaced by \mbox{\<^term>\<open>t*t\<close>}; before printing, every occurrence of \<^term>\<open>u*u\<close> is replaced by \mbox{\<^term>\<open>sq' u\<close>}. Internally, \<^const>\<open>sq'\<close> does not exist. This is the advantage of abbreviations over definitions: definitions need to be expanded explicitly (\autoref{sec:rewr-defs}) whereas abbreviations are already expanded upon parsing. However, abbreviations should be introduced sparingly: if abused, they can lead to a confusing discrepancy between the internal and external view of a term. The ASCII representation of \<open>\<equiv>\<close> is \texttt{==} or \xsymbol{equiv}. \subsection{Recursive Functions} \label{sec:recursive-funs} Recursive functions are defined with \indexed{\isacom{fun}}{fun} by pattern matching over datatype constructors. The order of equations matters, as in functional programming languages. However, all HOL functions must be total. This simplifies the logic --- terms are always defined --- but means that recursive functions must terminate. Otherwise one could define a function \<^prop>\<open>f n = f n + (1::nat)\<close> and conclude \mbox{\<^prop>\<open>(0::nat) = 1\<close>} by subtracting \<^term>\<open>f n\<close> on both sides. Isabelle's automatic termination checker requires that the arguments of recursive calls on the right-hand side must be strictly smaller than the arguments on the left-hand side. In the simplest case, this means that one fixed argument position decreases in size with each recursive call. The size is measured as the number of constructors (excluding 0-ary ones, e.g., \<open>Nil\<close>). Lexicographic combinations are also recognized. In more complicated situations, the user may have to prove termination by hand. For details see~@{cite Krauss}. Functions defined with \isacom{fun} come with their own induction schema that mirrors the recursion schema and is derived from the termination order. For example, \<close> fun div2 :: "nat \<Rightarrow> nat" where "div2 0 = 0" | "div2 (Suc 0) = 0" | "div2 (Suc(Suc n)) = Suc(div2 n)" text\<open>does not just define \<^const>\<open>div2\<close> but also proves a customized induction rule: \[ \inferrule{ \mbox{@{thm (prem 1) div2.induct}}\\ \mbox{@{thm (prem 2) div2.induct}}\\ \mbox{@{thm (prem 3) div2.induct}}} {\mbox{@{thm (concl) div2.induct[of _ "m"]}}} \] This customized induction rule can simplify inductive proofs. For example, \<close> lemma "div2(n) = n div 2" apply(induction n rule: div2.induct) txt\<open>(where the infix \<open>div\<close> is the predefined division operation) yields the subgoals @{subgoals[display,margin=65]} An application of \<open>auto\<close> finishes the proof. Had we used ordinary structural induction on \<open>n\<close>, the proof would have needed an additional case analysis in the induction step. This example leads to the following induction heuristic: \begin{quote} \emph{Let \<open>f\<close> be a recursive function. If the definition of \<open>f\<close> is more complicated than having one equation for each constructor of some datatype, then properties of \<open>f\<close> are best proved via \<open>f.induct\<close>.\index{inductionrule@\<open>.induct\<close>}} \end{quote} The general case is often called \concept{computation induction}, because the induction follows the (terminating!) computation. For every defining equation \begin{quote} \<open>f(e) = \<dots> f(r\<^sub>1) \<dots> f(r\<^sub>k) \<dots>\<close> \end{quote} where \<open>f(r\<^sub>i)\<close>, \<open>i=1\<dots>k\<close>, are all the recursive calls, the induction rule \<open>f.induct\<close> contains one premise of the form \begin{quote} \<open>P(r\<^sub>1) \<Longrightarrow> \<dots> \<Longrightarrow> P(r\<^sub>k) \<Longrightarrow> P(e)\<close> \end{quote} If \<open>f :: \<tau>\<^sub>1 \<Rightarrow> \<dots> \<Rightarrow> \<tau>\<^sub>n \<Rightarrow> \<tau>\<close> then \<open>f.induct\<close> is applied like this: \begin{quote} \isacom{apply}\<open>(induction x\<^sub>1 \<dots> x\<^sub>n rule: f.induct)\<close>\index{inductionrule@\<open>induction ... rule:\<close>} \end{quote} where typically there is a call \<open>f x\<^sub>1 \<dots> x\<^sub>n\<close> in the goal. But note that the induction rule does not mention \<open>f\<close> at all, except in its name, and is applicable independently of \<open>f\<close>. \subsection*{Exercises} \begin{exercise} Starting from the type \<open>'a tree\<close> defined in the text, define a function \<open>contents ::\<close> \<^typ>\<open>'a tree \<Rightarrow> 'a list\<close> that collects all values in a tree in a list, in any order, without removing duplicates. Then define a function \<open>sum_tree ::\<close> \<^typ>\<open>nat tree \<Rightarrow> nat\<close> that sums up all values in a tree of natural numbers and prove \<^prop>\<open>sum_tree t = sum_list(contents t)\<close> (where \<^const>\<open>sum_list\<close> is predefined). \end{exercise} \begin{exercise} Define the two functions \<open>pre_order\<close> and \<open>post_order\<close> of type @{typ "'a tree \<Rightarrow> 'a list"} that traverse a tree and collect all stored values in the respective order in a list. Prove \<^prop>\<open>pre_order (mirror t) = rev (post_order t)\<close>. \end{exercise} \begin{exercise} Define a function \<open>intersperse ::\<close> \<^typ>\<open>'a \<Rightarrow> 'a list \<Rightarrow> 'a list\<close> such that \<open>intersperse a [x\<^sub>1, ..., x\<^sub>n] = [x\<^sub>1, a, x\<^sub>2, a, ..., a, x\<^sub>n]\<close>. Now prove that \<^prop>\<open>map f (intersperse a xs) = intersperse (f a) (map f xs)\<close>. \end{exercise} \section{Induction Heuristics}\index{induction heuristics} We have already noted that theorems about recursive functions are proved by induction. In case the function has more than one argument, we have followed the following heuristic in the proofs about the append function: \begin{quote} \emph{Perform induction on argument number $i$\\ if the function is defined by recursion on argument number $i$.} \end{quote} The key heuristic, and the main point of this section, is to \emph{generalize the goal before induction}. The reason is simple: if the goal is too specific, the induction hypothesis is too weak to allow the induction step to go through. Let us illustrate the idea with an example. Function \<^const>\<open>rev\<close> has quadratic worst-case running time because it calls append for each element of the list and append is linear in its first argument. A linear time version of \<^const>\<open>rev\<close> requires an extra argument where the result is accumulated gradually, using only~\<open>#\<close>: \<close> (*<*) apply auto done (*>*) fun itrev :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where "itrev [] ys = ys" | "itrev (x#xs) ys = itrev xs (x#ys)" text\<open>The behaviour of \<^const>\<open>itrev\<close> is simple: it reverses its first argument by stacking its elements onto the second argument, and it returns that second argument when the first one becomes empty. Note that \<^const>\<open>itrev\<close> is tail-recursive: it can be compiled into a loop; no stack is necessary for executing it. Naturally, we would like to show that \<^const>\<open>itrev\<close> does indeed reverse its first argument provided the second one is empty: \<close> lemma "itrev xs [] = rev xs" txt\<open>There is no choice as to the induction variable: \<close> apply(induction xs) apply(auto) txt\<open> Unfortunately, this attempt does not prove the induction step: @{subgoals[display,margin=70]} The induction hypothesis is too weak. The fixed argument,~\<^term>\<open>[]\<close>, prevents it from rewriting the conclusion. This example suggests a heuristic: \begin{quote} \emph{Generalize goals for induction by replacing constants by variables.} \end{quote} Of course one cannot do this naively: \<^prop>\<open>itrev xs ys = rev xs\<close> is just not true. The correct generalization is \<close> (*<*)oops(*>*) lemma "itrev xs ys = rev xs @ ys" (*<*)apply(induction xs, auto)(*>*) txt\<open> If \<open>ys\<close> is replaced by \<^term>\<open>[]\<close>, the right-hand side simplifies to \<^term>\<open>rev xs\<close>, as required. In this instance it was easy to guess the right generalization. Other situations can require a good deal of creativity. Although we now have two variables, only \<open>xs\<close> is suitable for induction, and we repeat our proof attempt. Unfortunately, we are still not there: @{subgoals[display,margin=65]} The induction hypothesis is still too weak, but this time it takes no intuition to generalize: the problem is that the \<open>ys\<close> in the induction hypothesis is fixed, but the induction hypothesis needs to be applied with \<^term>\<open>a # ys\<close> instead of \<open>ys\<close>. Hence we prove the theorem for all \<open>ys\<close> instead of a fixed one. We can instruct induction to perform this generalization for us by adding \<open>arbitrary: ys\<close>\index{arbitrary@\<open>arbitrary:\<close>}. \<close> (*<*)oops lemma "itrev xs ys = rev xs @ ys" (*>*) apply(induction xs arbitrary: ys) txt\<open>The induction hypothesis in the induction step is now universally quantified over \<open>ys\<close>: @{subgoals[display,margin=65]} Thus the proof succeeds: \<close> apply auto done text\<open> This leads to another heuristic for generalization: \begin{quote} \emph{Generalize induction by generalizing all free variables\\ {\em(except the induction variable itself)}.} \end{quote} Generalization is best performed with \<open>arbitrary: y\<^sub>1 \<dots> y\<^sub>k\<close>. This heuristic prevents trivial failures like the one above. However, it should not be applied blindly. It is not always required, and the additional quantifiers can complicate matters in some cases. The variables that need to be quantified are typically those that change in recursive calls. \subsection*{Exercises} \begin{exercise} Write a tail-recursive variant of the \<open>add\<close> function on \<^typ>\<open>nat\<close>: \<^term>\<open>itadd :: nat \<Rightarrow> nat \<Rightarrow> nat\<close>. Tail-recursive means that in the recursive case, \<open>itadd\<close> needs to call itself directly: \mbox{\<^term>\<open>itadd (Suc m) n\<close>} \<open>= itadd \<dots>\<close>. Prove \<^prop>\<open>itadd m n = add m n\<close>. \end{exercise} \section{Simplification} So far we have talked a lot about simplifying terms without explaining the concept. \conceptidx{Simplification}{simplification} means \begin{itemize} \item using equations $l = r$ from left to right (only), \item as long as possible. \end{itemize} To emphasize the directionality, equations that have been given the \<open>simp\<close> attribute are called \conceptidx{simplification rules}{simplification rule}. Logically, they are still symmetric, but proofs by simplification use them only in the left-to-right direction. The proof tool that performs simplifications is called the \concept{simplifier}. It is the basis of \<open>auto\<close> and other related proof methods. The idea of simplification is best explained by an example. Given the simplification rules \[ \begin{array}{rcl@ {\quad}l} \<^term>\<open>0 + n::nat\<close> &\<open>=\<close>& \<open>n\<close> & (1) \\ \<^term>\<open>(Suc m) + n\<close> &\<open>=\<close>& \<^term>\<open>Suc (m + n)\<close> & (2) \\ \<open>(Suc m \<le> Suc n)\<close> &\<open>=\<close>& \<open>(m \<le> n)\<close> & (3)\\ \<open>(0 \<le> m)\<close> &\<open>=\<close>& \<^const>\<open>True\<close> & (4) \end{array} \] the formula \<^prop>\<open>0 + Suc 0 \<le> Suc 0 + x\<close> is simplified to \<^const>\<open>True\<close> as follows: \[ \begin{array}{r@ {}c@ {}l@ {\quad}l} \<open>(0 + Suc 0\<close> & \leq & \<open>Suc 0 + x)\<close> & \stackrel{(1)}{=} \\ \<open>(Suc 0\<close> & \leq & \<open>Suc 0 + x)\<close> & \stackrel{(2)}{=} \\ \<open>(Suc 0\<close> & \leq & \<open>Suc (0 + x))\<close> & \stackrel{(3)}{=} \\ \<open>(0\<close> & \leq & \<open>0 + x)\<close> & \stackrel{(4)}{=} \\[1ex] & \<^const>\<open>True\<close> \end{array} \] Simplification is often also called \concept{rewriting} and simplification rules \conceptidx{rewrite rules}{rewrite rule}. \subsection{Simplification Rules} The attribute \<open>simp\<close> declares theorems to be simplification rules, which the simplifier will use automatically. In addition, \isacom{datatype} and \isacom{fun} commands implicitly declare some simplification rules: \isacom{datatype} the distinctness and injectivity rules, \isacom{fun} the defining equations. Definitions are not declared as simplification rules automatically! Nearly any theorem can become a simplification rule. The simplifier will try to transform it into an equation. For example, the theorem \<^prop>\<open>\<not> P\<close> is turned into \<^prop>\<open>P = False\<close>. Only equations that really simplify, like \<^prop>\<open>rev (rev xs) = xs\<close> and \<^prop>\<open>xs @ [] = xs\<close>, should be declared as simplification rules. Equations that may be counterproductive as simplification rules should only be used in specific proof steps (see \autoref{sec:simp} below). Distributivity laws, for example, alter the structure of terms and can produce an exponential blow-up. \subsection{Conditional Simplification Rules} Simplification rules can be conditional. Before applying such a rule, the simplifier will first try to prove the preconditions, again by simplification. For example, given the simplification rules \begin{quote} \<^prop>\<open>p(0::nat) = True\<close>\\ \<^prop>\<open>p(x) \<Longrightarrow> f(x) = g(x)\<close>, \end{quote} the term \<^term>\<open>f(0::nat)\<close> simplifies to \<^term>\<open>g(0::nat)\<close> but \<^term>\<open>f(1::nat)\<close> does not simplify because \<^prop>\<open>p(1::nat)\<close> is not provable. \subsection{Termination} Simplification can run forever, for example if both \<^prop>\<open>f x = g x\<close> and \<^prop>\<open>g(x) = f(x)\<close> are simplification rules. It is the user's responsibility not to include simplification rules that can lead to nontermination, either on their own or in combination with other simplification rules. The right-hand side of a simplification rule should always be ``simpler'' than the left-hand side --- in some sense. But since termination is undecidable, such a check cannot be automated completely and Isabelle makes little attempt to detect nontermination. When conditional simplification rules are applied, their preconditions are proved first. Hence all preconditions need to be simpler than the left-hand side of the conclusion. For example \begin{quote} \<^prop>\<open>n < m \<Longrightarrow> (n < Suc m) = True\<close> \end{quote} is suitable as a simplification rule: both \<^prop>\<open>n<m\<close> and \<^const>\<open>True\<close> are simpler than \mbox{\<^prop>\<open>n < Suc m\<close>}. But \begin{quote} \<^prop>\<open>Suc n < m \<Longrightarrow> (n < m) = True\<close> \end{quote} leads to nontermination: when trying to rewrite \<^prop>\<open>n<m\<close> to \<^const>\<open>True\<close> one first has to prove \mbox{\<^prop>\<open>Suc n < m\<close>}, which can be rewritten to \<^const>\<open>True\<close> provided \<^prop>\<open>Suc(Suc n) < m\<close>, \emph{ad infinitum}. \subsection{The \indexed{\<open>simp\<close>}{simp} Proof Method} \label{sec:simp} So far we have only used the proof method \<open>auto\<close>. Method \<open>simp\<close> is the key component of \<open>auto\<close>, but \<open>auto\<close> can do much more. In some cases, \<open>auto\<close> is overeager and modifies the proof state too much. In such cases the more predictable \<open>simp\<close> method should be used. Given a goal \begin{quote} \<open>1. \<lbrakk> P\<^sub>1; \<dots>; P\<^sub>m \<rbrakk> \<Longrightarrow> C\<close> \end{quote} the command \begin{quote} \isacom{apply}\<open>(simp add: th\<^sub>1 \<dots> th\<^sub>n)\<close> \end{quote} simplifies the assumptions \<open>P\<^sub>i\<close> and the conclusion \<open>C\<close> using \begin{itemize} \item all simplification rules, including the ones coming from \isacom{datatype} and \isacom{fun}, \item the additional lemmas \<open>th\<^sub>1 \<dots> th\<^sub>n\<close>, and \item the assumptions. \end{itemize} In addition to or instead of \<open>add\<close> there is also \<open>del\<close> for removing simplification rules temporarily. Both are optional. Method \<open>auto\<close> can be modified similarly: \begin{quote} \isacom{apply}\<open>(auto simp add: \<dots> simp del: \<dots>)\<close> \end{quote} Here the modifiers are \<open>simp add\<close> and \<open>simp del\<close> instead of just \<open>add\<close> and \<open>del\<close> because \<open>auto\<close> does not just perform simplification. Note that \<open>simp\<close> acts only on subgoal 1, \<open>auto\<close> acts on all subgoals. There is also \<open>simp_all\<close>, which applies \<open>simp\<close> to all subgoals. \subsection{Rewriting with Definitions} \label{sec:rewr-defs} Definitions introduced by the command \isacom{definition} can also be used as simplification rules, but by default they are not: the simplifier does not expand them automatically. Definitions are intended for introducing abstract concepts and not merely as abbreviations. Of course, we need to expand the definition initially, but once we have proved enough abstract properties of the new constant, we can forget its original definition. This style makes proofs more robust: if the definition has to be changed, only the proofs of the abstract properties will be affected. The definition of a function \<open>f\<close> is a theorem named \<open>f_def\<close> and can be added to a call of \<open>simp\<close> like any other theorem: \begin{quote} \isacom{apply}\<open>(simp add: f_def)\<close> \end{quote} In particular, let-expressions can be unfolded by making @{thm[source] Let_def} a simplification rule. \subsection{Case Splitting With \<open>simp\<close>} Goals containing if-expressions are automatically split into two cases by \<open>simp\<close> using the rule \begin{quote} \<^prop>\<open>P(if A then s else t) = ((A \<longrightarrow> P(s)) \<and> (~A \<longrightarrow> P(t)))\<close> \end{quote} For example, \<open>simp\<close> can prove \begin{quote} \<^prop>\<open>(A \<and> B) = (if A then B else False)\<close> \end{quote} because both \<^prop>\<open>A \<longrightarrow> (A & B) = B\<close> and \<^prop>\<open>~A \<longrightarrow> (A & B) = False\<close> simplify to \<^const>\<open>True\<close>. We can split case-expressions similarly. For \<open>nat\<close> the rule looks like this: @{prop[display,margin=65,indent=4]"P(case e of 0 \<Rightarrow> a | Suc n \<Rightarrow> b n) = ((e = 0 \<longrightarrow> P a) \<and> (\<forall>n. e = Suc n \<longrightarrow> P(b n)))"} Case expressions are not split automatically by \<open>simp\<close>, but \<open>simp\<close> can be instructed to do so: \begin{quote} \isacom{apply}\<open>(simp split: nat.split)\<close> \end{quote} splits all case-expressions over natural numbers. For an arbitrary datatype \<open>t\<close> it is \<open>t.split\<close>\index{split@\<open>.split\<close>} instead of @{thm[source] nat.split}. Method \<open>auto\<close> can be modified in exactly the same way. The modifier \indexed{\<open>split:\<close>}{split} can be followed by multiple names. Splitting if or case-expressions in the assumptions requires \<open>split: if_splits\<close> or \<open>split: t.splits\<close>. \ifsem\else \subsection{Rewriting \<open>let\<close> and Numerals} Let-expressions (\<^term>\<open>let x = s in t\<close>) can be expanded explicitly with the simplification rule @{thm[source] Let_def}. The simplifier will not expand \<open>let\<close>s automatically in many cases. Numerals of type \<^typ>\<open>nat\<close> can be converted to \<^const>\<open>Suc\<close> terms with the simplification rule @{thm[source] numeral_eq_Suc}. This is required, for example, when a function that is defined by pattern matching with \<^const>\<open>Suc\<close> is applied to a numeral: if \<open>f\<close> is defined by \<open>f 0 = ...\<close> and \<open>f (Suc n) = ...\<close>, the simplifier cannot simplify \<open>f 2\<close> unless \<open>2\<close> is converted to \<^term>\<open>Suc(Suc 0)\<close> (via @{thm[source] numeral_eq_Suc}). \fi \subsection*{Exercises} \exercise\label{exe:tree0} Define a datatype \<open>tree0\<close> of binary tree skeletons which do not store any information, neither in the inner nodes nor in the leaves. Define a function \<open>nodes :: tree0 \<Rightarrow> nat\<close> that counts the number of all nodes (inner nodes and leaves) in such a tree. Consider the following recursive function: \<close> (*<*) datatype tree0 = Tip | Node tree0 tree0 (*>*) fun explode :: "nat \<Rightarrow> tree0 \<Rightarrow> tree0" where "explode 0 t = t" | "explode (Suc n) t = explode n (Node t t)" text \<open> Find an equation expressing the size of a tree after exploding it (\noquotes{@{term [source] "nodes (explode n t)"}}) as a function of \<^term>\<open>nodes t\<close> and \<open>n\<close>. Prove your equation. You may use the usual arithmetic operators, including the exponentiation operator ``\<open>^\<close>''. For example, \noquotes{@{prop [source] "2 ^ 2 = 4"}}. Hint: simplifying with the list of theorems @{thm[source] algebra_simps} takes care of common algebraic properties of the arithmetic operators. \endexercise \exercise Define arithmetic expressions in one variable over integers (type \<^typ>\<open>int\<close>) as a data type: \<close> datatype exp = Var | Const int | Add exp exp | Mult exp exp text\<open> Define a function \noquotes{@{term [source]"eval :: exp \<Rightarrow> int \<Rightarrow> int"}} such that \<^term>\<open>eval e x\<close> evaluates \<open>e\<close> at the value \<open>x\<close>. A polynomial can be represented as a list of coefficients, starting with the constant. For example, \<^term>\<open>[4, 2, -1, 3::int]\<close> represents the polynomial $4 + 2x - x^2 + 3x^3$. Define a function \noquotes{@{term [source] "evalp :: int list \<Rightarrow> int \<Rightarrow> int"}} that evaluates a polynomial at the given value. Define a function \noquotes{@{term[source] "coeffs :: exp \<Rightarrow> int list"}} that transforms an expression into a polynomial. This may require auxiliary functions. Prove that \<open>coeffs\<close> preserves the value of the expression: \mbox{\<^prop>\<open>evalp (coeffs e) x = eval e x\<close>.} Hint: consider the hint in Exercise~\ref{exe:tree0}. \endexercise \<close> (*<*) end (*>*)