diff --git a/thys/Huffman/Huffman.thy b/thys/Huffman/Huffman.thy --- a/thys/Huffman/Huffman.thy +++ b/thys/Huffman/Huffman.thy @@ -1,2413 +1,2413 @@ (* Title: An Isabelle/HOL Formalization of the Textbook Proof of Huffman's Algorithm Author: Jasmin Christian Blanchette , 2008 Maintainer: Jasmin Christian Blanchette *) (*<*) theory Huffman imports Main begin (*>*) section \Introduction\ subsection \Binary Codes \label{binary-codes}\ text \ Suppose we want to encode strings over a finite source alphabet to sequences of bits. The approach used by ASCII and most other charsets is to map each source symbol to a distinct $k$-bit code word, where $k$ is fixed and is typically 8 or 16. To encode a string of symbols, we simply encode each symbol in turn. Decoding involves mapping each $k$-bit block back to the symbol it represents. Fixed-length codes are simple and fast, but they generally waste space. If we know the frequency $w_a$ of each source symbol $a$, we can save space by using shorter code words for the most frequent symbols. We say that a (variable-length) code is {\sl optimum\/} if it minimizes the sum $\sum_a w_a \vthinspace\delta_a$, where $\delta_a$ is the length of the binary code word for $a$. Information theory tells us that a code is optimum if for each source symbol $c$ the code word representing $c$ has length $$\textstyle \delta_c = \log_2 {1 \over p_c}, \qquad \hbox{where}\enskip p_c = {w_c \over \sum_a w_a}.$$ This number is generally not an integer, so we cannot use it directly. Nonetheless, the above criterion is a useful yardstick and paves the way for -arithmetic coding \<^cite>\"rissanen-1976"\, a generalization of the method +arithmetic coding \cite{rissanen-1976}, a generalization of the method presented here. \def\xabacabad{\xa\xb\xa\xc\xa\xb\xa\xd}% As an example, consider the source string `$\xabacabad$'. We have $$p_{\xa} = \tfrac{1}{2},\,\; p_{\xb} = \tfrac{1}{4},\,\; p_{\xc} = \tfrac{1}{8},\,\; p_{\xd} = \tfrac{1}{8}.$$ The optimum lengths for the binary code words are all integers, namely $$\delta_{\xa} = 1,\,\; \delta_{\xb} = 2,\,\; \delta_{\xc} = 3,\,\; \delta_{\xd} = 3,$$ and they are realized by the code $$C_1 = \{ \xa \mapsto 0,\, \xb \mapsto 10,\, \xc \mapsto 110,\, \xd \mapsto 111 \}.$$ Encoding `$\xabacabad$' produces the 14-bit code word 01001100100111. The code $C_1$ is optimum: No code that unambiguously encodes source symbols one at a time could do better than $C_1$ on the input `$\xa\xb\xa\xc\xa\xb\xa\xd$'. In particular, with a fixed-length code such as $$C_2 = \{ \xa \mapsto 00,\, \xb \mapsto 01,\, \xc \mapsto 10,\, \xd \mapsto 11 \}$$ we need at least 16~bits to encode `$\xabacabad$'. \ subsection \Binary Trees\ text \ Inside a program, binary codes can be represented by binary trees. For example, the trees\strut $$\vcenter{\hbox{\includegraphics[scale=1.25]{tree-abcd-unbalanced.pdf}}} \qquad \hbox{and} \qquad \vcenter{\hbox{\includegraphics[scale=1.25]{tree-abcd-balanced.pdf}}}$$ correspond to $C_1$ and $C_2$. The code word for a given symbol can be obtained as follows: Start at the root and descend toward the leaf node associated with the symbol one node at a time; generate a 0 whenever the left child of the current node is chosen and a 1 whenever the right child is chosen. The generated sequence of 0s and 1s is the code word. To avoid ambiguities, we require that only leaf nodes are labeled with symbols. This ensures that no code word is a prefix of another, thereby eliminating the source of all ambiguities.% \footnote{Strictly speaking, there is another potential source of ambiguity. If the alphabet consists of a single symbol $a$, that symbol could be mapped to the empty code word, and then any string $aa\ldots a$ would map to the empty bit sequence, giving the decoder no way to recover the original string's length. This scenario can be ruled out by requiring that the alphabet has cardinality 2 or more.} Codes that have this property are called {\sl prefix codes}. As an example of a code that does not have this property, consider the code associated with the tree\strut $$\vcenter{\hbox{\includegraphics[scale=1.25]{tree-abcd-non-prefix.pdf}}}$$ and observe that `$\xb\xb\xb$', `$\xb\xd$', and `$\xd\xb$' all map to the code word 111. Each node in a code tree is assigned a {\sl weight}. For a leaf node, the weight is the frequency of its symbol; for an inner node, it is the sum of the weights of its subtrees. Code trees can be annotated with their weights:\strut $$\vcenter{\hbox{\includegraphics[scale=1.25]% {tree-abcd-unbalanced-weighted.pdf}}} \qquad\qquad \vcenter{\hbox{\includegraphics[scale=1.25]% {tree-abcd-balanced-weighted.pdf}}}$$ For our purposes, it is sufficient to consider only full binary trees (trees whose inner nodes all have two children). This is because any inner node with only one child can advantageously be eliminated; for example,\strut $$\vcenter{\hbox{\includegraphics[scale=1.25]{tree-abc-non-full.pdf}}} \qquad \hbox{becomes} \qquad \vcenter{\hbox{\includegraphics[scale=1.25]{tree-abc-full.pdf}}}$$ \ subsection \Huffman's Algorithm\ text \ -David Huffman \<^cite>\"huffman-1952"\ discovered a simple algorithm for +David Huffman \cite{huffman-1952} discovered a simple algorithm for constructing an optimum code tree for specified symbol frequencies: Create a forest consisting of only leaf nodes, one for each symbol in the alphabet, taking the given symbol frequencies as initial weights for the nodes. Then pick the two trees $$\vcenter{\hbox{\includegraphics[scale=1.25]{tree-w1.pdf}}} \qquad \hbox{and} \qquad \vcenter{\hbox{\includegraphics[scale=1.25]{tree-w2.pdf}}}$$ \noindent\strut with the lowest weights and replace them with the tree $$\vcenter{\hbox{\includegraphics[scale=1.25]{tree-w1-w2.pdf}}}$$ Repeat this process until only one tree is left. As an illustration, executing the algorithm for the frequencies $$f_{\xd} = 3,\,\; f_{\xe} = 11,\,\; f_{\xf} = 5,\,\; f_{\xs} = 7,\,\; f_{\xz} = 2$$ gives rise to the following sequence of states:\strut \def\myscale{1}% \setbox\myboxi=\hbox{(9)\strut}% \setbox\myboxii=\hbox{\includegraphics[scale=\myscale]{tree-prime-step1.pdf}}% \setbox\myboxiii=\hbox{\includegraphics[scale=\myscale]{tree-prime-step2.pdf}}% $$(1)\quad\lower\ht\myboxii\hbox{\raise\ht\myboxi\box\myboxii} \qquad\qquad (2)\enskip\lower\ht\myboxiii\hbox{\raise\ht\myboxi\box\myboxiii}$$ \vskip.5\smallskipamount \noindent \setbox\myboxii=\hbox{\includegraphics[scale=\myscale]{tree-prime-step3.pdf}}% \setbox\myboxiii=\hbox{\includegraphics[scale=\myscale]{tree-prime-step4.pdf}}% \setbox\myboxiv=\hbox{\includegraphics[scale=\myscale]{tree-prime-step5.pdf}}% (3)\quad\lower\ht\myboxii\hbox{\raise\ht\myboxi\box\myboxii}\hfill\quad (4)\quad\lower\ht\myboxiii\hbox{\raise\ht\myboxi\box\myboxiii}\hfill (5)\enskip\lower\ht\myboxiv\hbox{\raise\ht\myboxi\box\myboxiv\,} \smallskip \noindent Tree~(5) is an optimum tree for the given frequencies. \ subsection \The Textbook Proof\ text \ Why does the algorithm work? In his article, Huffman gave some motivation but no real proof. For a proof sketch, we turn to Donald Knuth -\<^cite>\\p.~403--404\ in "knuth-1997"\: +\cite[p.~403--404]{knuth-1997}: \begin{quote} It is not hard to prove that this method does in fact minimize the weighted path length (i.e., $\sum_a w_a \vthinspace\delta_a$), by induction on $m$. Suppose we have $w_1 \le w_2 \le w_3 \le \cdots \le w_m$, where $m \ge 2$, and suppose that we are given a tree that minimizes the weighted path length. (Such a tree certainly exists, since only finitely many binary trees with $m$ terminal nodes are possible.) Let $V$ be an internal node of maximum distance from the root. If $w_1$ and $w_2$ are not the weights already attached to the children of $V$, we can interchange them with the values that are already there; such an interchange does not increase the weighted path length. Thus there is a tree that minimizes the weighted path length and contains the subtree\strut $$\vcenter{\hbox{\includegraphics[scale=1.25]{tree-w1-w2-leaves.pdf}}}$$ Now it is easy to prove that the weighted path length of such a tree is minimized if and only if the tree with $$\vcenter{\hbox{\includegraphics[scale=1.25]{tree-w1-w2-leaves.pdf}}} \qquad \hbox{replaced by} \qquad \vcenter{\hbox{\includegraphics[scale=1.25]{tree-w1-plus-w2.pdf}}}$$ has minimum path length for the weights $w_1 + w_2$, $w_3$, $\ldots\,$, $w_m$. \end{quote} \noindent There is, however, a small oddity in this proof: It is not clear why we must assert the existence of an optimum tree that contains the subtree $$\vcenter{\hbox{\includegraphics[scale=1.25]{tree-w1-w2-leaves.pdf}}}$$ Indeed, the formalization works without it. -Cormen et al.\ \<^cite>\\p.~385--391\ in "cormen-et-al-2001"\ provide a very similar +Cormen et al.\ \cite[p.~385--391]{cormen-et-al-2001} provide a very similar proof, articulated around the following propositions: \begin{quote} \textsl{\textbf{Lemma 16.2}} \\ Let $C$ be an alphabet in which each character $c \in C$ has frequency $f[c]$. Let $x$ and $y$ be two characters in $C$ having the lowest frequencies. Then there exists an optimal prefix code for $C$ in which the codewords for $x$ and $y$ have the same length and differ only in the last bit. \medskip \textsl{\textbf{Lemma 16.3}} \\ Let $C$ be a given alphabet with frequency $f[c]$ defined for each character $c \in C$. Let $x$ and $y$ be two characters in $C$ with minimum frequency. Let $C'$ be the alphabet $C$ with characters $x$, $y$ removed and (new) character $z$ added, so that $C' = C - \{x, y\} \cup {\{z\}}$; define $f$ for $C'$ as for $C$, except that $f[z] = f[x] + f[y]$. Let $T'$ be any tree representing an optimal prefix code for the alphabet $C'$. Then the tree $T$, obtained from $T'$ by replacing the leaf node for $z$ with an internal node having $x$ and $y$ as children, represents an optimal prefix code for the alphabet $C$. \medskip \textsl{\textbf{Theorem 16.4}} \\ Procedure \textsc{Huffman} produces an optimal prefix code. \end{quote} \ subsection \Overview of the Formalization\ text \ This document presents a formalization of the proof of Huffman's algorithm -written using Isabelle/HOL \<^cite>\"nipkow-et-al-2008"\. Our proof is based on the +written using Isabelle/HOL \cite{nipkow-et-al-2008}. Our proof is based on the informal proofs given by Knuth and Cormen et al. The development was done -independently of Laurent Th\'ery's Coq proof \<^cite>\"thery-2003" and "thery-2004"\, +independently of Laurent Th\'ery's Coq proof \cite{thery-2003,thery-2004}, which through its ``cover'' concept represents a considerable departure from the textbook proof. The development consists of a little under 100 lemmas and theorems. Most of them have very short proofs thanks to the extensive use of simplification rules and custom induction rules. The remaining proofs are written using the -structured proof format Isar \<^cite>\"wenzel-2008"\. +structured proof format Isar \cite{wenzel-2008}. \ subsection \Head of the Theory File\ text \ The Isabelle theory starts in the standard way. \myskip \noindent \isacommand{theory} \Huffman\ \\ \isacommand{imports} \Main\ \\ \isacommand{begin} \myskip \noindent We attach the \simp\ attribute to some predefined lemmas to add them to the default set of simplification rules. \ declare Int_Un_distrib [simp] Int_Un_distrib2 [simp] max.absorb1 [simp] max.absorb2 [simp] section \Definition of Prefix Code Trees and Forests \label{trees-and-forests}\ subsection \Tree Type\ text \ A {\sl prefix code tree\/} is a full binary tree in which leaf nodes are of the form @{term "Leaf w a"}, where @{term a} is a symbol and @{term w} is the frequency associated with @{term a}, and inner nodes are of the form @{term "Node w t\<^sub>1 t\<^sub>2"}, where @{term t\<^sub>1} and @{term t\<^sub>2} are the left and right subtrees and @{term w} caches the sum of the weights of @{term t\<^sub>1} and @{term t\<^sub>2}. Prefix code trees are polymorphic on the symbol datatype~@{typ 'a}. \ datatype 'a tree = Leaf nat 'a | Node nat "('a tree)" "('a tree)" subsection \Forest Type\ text \ The intermediate steps of Huffman's algorithm involve a list of prefix code trees, or {\sl prefix code forest}. \ type_synonym 'a forest = "'a tree list" subsection \Alphabet\ text \ The {\sl alphabet\/} of a code tree is the set of symbols appearing in the tree's leaf nodes. \ primrec alphabet :: "'a tree \ 'a set" where "alphabet (Leaf w a) = {a}" | "alphabet (Node w t\<^sub>1 t\<^sub>2) = alphabet t\<^sub>1 \ alphabet t\<^sub>2" text \ For sets and predicates, Isabelle gives us the choice between inductive definitions (\isakeyword{inductive\_set} and \isakeyword{inductive}) and recursive functions (\isakeyword{primrec}, \isakeyword{fun}, and \isakeyword{function}). In this development, we consistently favor recursion over induction, for two reasons: \begin{myitemize} \item Recursion gives rise to simplification rules that greatly help automatic proof tactics. In contrast, reasoning about inductively defined sets and predicates involves introduction and elimination rules, which are more clumsy than simplification rules. \item Isabelle's counterexample generator \isakeyword{quickcheck} -\<^cite>\"berghofer-nipkow-2004"\, which we used extensively during the top-down +\cite{berghofer-nipkow-2004}, which we used extensively during the top-down development of the proof (together with \isakeyword{sorry}), has better support for recursive definitions. \end{myitemize} The alphabet of a forest is defined as the union of the alphabets of the trees that compose it. Although Isabelle supports overloading for non-overlapping types, we avoid many type inference problems by attaching an `\raise.3ex\hbox{\\<^sub>F\}' subscript to the forest generalizations of functions defined on trees. \ primrec alphabet\<^sub>F :: "'a forest \ 'a set" where "alphabet\<^sub>F [] = {}" | "alphabet\<^sub>F (t # ts) = alphabet t \ alphabet\<^sub>F ts" text \ Alphabets are central to our proofs, and we need the following basic facts about them. \ lemma finite_alphabet[simp]: "finite (alphabet t)" by (induct t) auto lemma exists_in_alphabet: "\a. a \ alphabet t" by (induct t) auto subsection \Consistency\ text \ A tree is {\sl consistent\/} if for each inner node the alphabets of the two subtrees are disjoint. Intuitively, this means that every symbol in the alphabet occurs in exactly one leaf node. Consistency is a sufficient condition for $\delta_a$ (the length of the {\sl unique\/} code word for $a$) to be defined. Although this well\-formed\-ness property is not mentioned in algorithms -textbooks \<^cite>\"aho-et-al-1983" and "cormen-et-al-2001" and "knuth-1997"\, it is essential +textbooks \cite{aho-et-al-1983,cormen-et-al-2001,knuth-1997}, it is essential and appears as an assumption in many of our lemmas. \ primrec consistent :: "'a tree \ bool" where "consistent (Leaf w a) = True" | "consistent (Node w t\<^sub>1 t\<^sub>2) = (alphabet t\<^sub>1 \ alphabet t\<^sub>2 = {} \ consistent t\<^sub>1 \ consistent t\<^sub>2)" primrec consistent\<^sub>F :: "'a forest \ bool" where "consistent\<^sub>F [] = True" | "consistent\<^sub>F (t # ts) = (alphabet t \ alphabet\<^sub>F ts = {} \ consistent t \ consistent\<^sub>F ts)" text \ Several of our proofs are by structural induction on consistent trees $t$ and involve one symbol $a$. These proofs typically distinguish the following cases. \begin{myitemize} \item[] {\sc Base case:}\enspace $t = @{term "Leaf w b"}$. \item[] {\sc Induction step:}\enspace $t = @{term "Node w t\<^sub>1 t\<^sub>2"}$. \item[] \noindent\kern\leftmargin {\sc Subcase 1:}\enspace $a$ belongs to @{term t\<^sub>1} but not to @{term t\<^sub>2}. \item[] \noindent\kern\leftmargin {\sc Subcase 2:}\enspace $a$ belongs to @{term t\<^sub>2} but not to @{term t\<^sub>1}. \item[] \noindent\kern\leftmargin {\sc Subcase 3:}\enspace $a$ belongs to neither @{term t\<^sub>1} nor @{term t\<^sub>2}. \end{myitemize} \noindent Thanks to the consistency assumption, we can rule out the subcase where $a$ belongs to both subtrees. Instead of performing the above case distinction manually, we encode it in a custom induction rule. This saves us from writing repetitive proof scripts and helps Isabelle's automatic proof tactics. \ lemma tree_induct_consistent[consumes 1, case_names base step\<^sub>1 step\<^sub>2 step\<^sub>3]: "\consistent t; \w\<^sub>b b a. P (Leaf w\<^sub>b b) a; \w t\<^sub>1 t\<^sub>2 a. \consistent t\<^sub>1; consistent t\<^sub>2; alphabet t\<^sub>1 \ alphabet t\<^sub>2 = {}; a \ alphabet t\<^sub>1; a \ alphabet t\<^sub>2; P t\<^sub>1 a; P t\<^sub>2 a\ \ P (Node w t\<^sub>1 t\<^sub>2) a; \w t\<^sub>1 t\<^sub>2 a. \consistent t\<^sub>1; consistent t\<^sub>2; alphabet t\<^sub>1 \ alphabet t\<^sub>2 = {}; a \ alphabet t\<^sub>1; a \ alphabet t\<^sub>2; P t\<^sub>1 a; P t\<^sub>2 a\ \ P (Node w t\<^sub>1 t\<^sub>2) a; \w t\<^sub>1 t\<^sub>2 a. \consistent t\<^sub>1; consistent t\<^sub>2; alphabet t\<^sub>1 \ alphabet t\<^sub>2 = {}; a \ alphabet t\<^sub>1; a \ alphabet t\<^sub>2; P t\<^sub>1 a; P t\<^sub>2 a\ \ P (Node w t\<^sub>1 t\<^sub>2) a\ \ P t a" txt \ The proof relies on the \textit{induction\_schema} and \textit{lexicographic\_order} tactics, which automate the most tedious aspects of deriving induction rules. The alternative would have been to perform a standard structural induction on @{term t} and proceed by cases, which is straightforward but long-winded. \ apply rotate_tac apply induction_schema apply atomize_elim apply (case_tac t) apply fastforce apply fastforce by lexicographic_order text \ The \textit{induction\_schema} tactic reduces the putative induction rule to simpler proof obligations. Internally, it reuses the machinery that constructs the default induction rules. The resulting proof obligations concern (a)~case completeness, (b)~invariant preservation (in our case, tree consistency), and (c)~wellfoundedness. For @{thm [source] tree_induct_consistent}, the obligations (a)~and (b) can be discharged using Isabelle's simplifier and classical reasoner, whereas (c) requires a single invocation of \textit{lexicographic\_order}, a tactic that was originally designed to prove termination of recursive functions -\<^cite>\"bulwahn-et-al-2007" and "krauss-2007" and "krauss-2009"\. +\cite{bulwahn-et-al-2007,krauss-2007,krauss-2009}. \ subsection \Symbol Depths\ text \ The {\sl depth\/} of a symbol (which we denoted by $\delta_a$ in Section~\ref{binary-codes}) is the length of the path from the root to the leaf node labeled with that symbol, or equivalently the length of the code word for the symbol. Symbols that do not occur in the tree or that occur at the root of a one-node tree have depth 0. If a symbol occurs in several leaf nodes (which may happen with inconsistent trees), the depth is arbitrarily defined in terms of the leftmost node labeled with that symbol. \ primrec depth :: "'a tree \ 'a \ nat" where "depth (Leaf w b) a = 0" | "depth (Node w t\<^sub>1 t\<^sub>2) a = (if a \ alphabet t\<^sub>1 then depth t\<^sub>1 a + 1 else if a \ alphabet t\<^sub>2 then depth t\<^sub>2 a + 1 else 0)" text \ The definition may seem very inefficient from a functional programming point of view, but it does not matter, because unlike Huffman's algorithm, the @{const depth} function is merely a reasoning tool and is never actually executed. \ subsection \Height\ text \ The {\sl height\/} of a tree is the length of the longest path from the root to a leaf node, or equivalently the length of the longest code word. This is readily generalized to forests by taking the maximum of the trees' heights. Note that a tree has height 0 if and only if it is a leaf node, and that a forest has height 0 if and only if all its trees are leaf nodes. \ primrec height :: "'a tree \ nat" where "height (Leaf w a) = 0" | "height (Node w t\<^sub>1 t\<^sub>2) = max (height t\<^sub>1) (height t\<^sub>2) + 1" primrec height\<^sub>F :: "'a forest \ nat" where "height\<^sub>F [] = 0" | "height\<^sub>F (t # ts) = max (height t) (height\<^sub>F ts)" text \ The depth of any symbol in the tree is bounded by the tree's height, and there exists a symbol with a depth equal to the height. \ lemma depth_le_height: "depth t a \ height t" by (induct t) auto lemma exists_at_height: "consistent t \ \a \ alphabet t. depth t a = height t" proof (induct t) case Leaf thus ?case by simp next case (Node w t\<^sub>1 t\<^sub>2) note hyps = Node let ?t = "Node w t\<^sub>1 t\<^sub>2" from hyps obtain b where b: "b \ alphabet t\<^sub>1" "depth t\<^sub>1 b = height t\<^sub>1" by auto from hyps obtain c where c: "c \ alphabet t\<^sub>2" "depth t\<^sub>2 c = height t\<^sub>2" by auto let ?a = "if height t\<^sub>1 \ height t\<^sub>2 then b else c" from b c have "?a \ alphabet ?t" "depth ?t ?a = height ?t" using \consistent ?t\ by auto thus "\a \ alphabet ?t. depth ?t a = height ?t" .. qed text \ The following elimination rules help Isabelle's classical prover, notably the \textit{auto} tactic. They are easy consequences of the inequation @{thm depth_le_height[no_vars]}. \ lemma depth_max_heightE_left[elim!]: "\depth t\<^sub>1 a = max (height t\<^sub>1) (height t\<^sub>2); \depth t\<^sub>1 a = height t\<^sub>1; height t\<^sub>1 \ height t\<^sub>2\ \ P\ \ P" by (cut_tac t = t\<^sub>1 and a = a in depth_le_height) simp lemma depth_max_heightE_right[elim!]: "\depth t\<^sub>2 a = max (height t\<^sub>1) (height t\<^sub>2); \depth t\<^sub>2 a = height t\<^sub>2; height t\<^sub>2 \ height t\<^sub>1\ \ P\ \ P" by (cut_tac t = t\<^sub>2 and a = a in depth_le_height) simp text \ We also need the following lemma. \ lemma height_gt_0_alphabet_eq_imp_height_gt_0: assumes "height t > 0" "consistent t" "alphabet t = alphabet u" shows "height u > 0" proof (cases t) case Leaf thus ?thesis using assms by simp next case (Node w t\<^sub>1 t\<^sub>2) note t = Node from exists_in_alphabet obtain b where b: "b \ alphabet t\<^sub>1" .. from exists_in_alphabet obtain c where c: "c \ alphabet t\<^sub>2" .. from b c have bc: "b \ c" using t \consistent t\ by fastforce show ?thesis proof (cases u) case Leaf thus ?thesis using b c bc t assms by auto next case Node thus ?thesis by simp qed qed subsection \Symbol Frequencies\ text \ The {\sl frequency\/} of a symbol (which we denoted by $w_a$ in Section~\ref{binary-codes}) is the sum of the weights attached to the leaf nodes labeled with that symbol. If the tree is consistent, the sum comprises at most one nonzero term. The frequency is then the weight of the leaf node labeled with the symbol, or 0 if there is no such node. The generalization to forests is straightforward. \ primrec freq :: "'a tree \ 'a \ nat" where "freq (Leaf w a) b = (if b = a then w else 0)" | "freq (Node w t\<^sub>1 t\<^sub>2) b = freq t\<^sub>1 b + freq t\<^sub>2 b" primrec freq\<^sub>F :: "'a forest \ 'a \ nat" where "freq\<^sub>F [] b = 0" | "freq\<^sub>F (t # ts) b = freq t b + freq\<^sub>F ts b" text \ Alphabet and symbol frequencies are intimately related. Simplification rules ensure that sums of the form @{term "freq t\<^sub>1 a + freq t\<^sub>2 a"} collapse to a single term when we know which tree @{term a} belongs to. \ lemma notin_alphabet_imp_freq_0[simp]: "a \ alphabet t \ freq t a = 0" by (induct t) simp+ lemma notin_alphabet\<^sub>F_imp_freq\<^sub>F_0[simp]: "a \ alphabet\<^sub>F ts \ freq\<^sub>F ts a = 0" by (induct ts) simp+ lemma freq_0_right[simp]: "\alphabet t\<^sub>1 \ alphabet t\<^sub>2 = {}; a \ alphabet t\<^sub>1\ \ freq t\<^sub>2 a = 0" by (auto intro: notin_alphabet_imp_freq_0 simp: disjoint_iff_not_equal) lemma freq_0_left[simp]: "\alphabet t\<^sub>1 \ alphabet t\<^sub>2 = {}; a \ alphabet t\<^sub>2\ \ freq t\<^sub>1 a = 0" by (auto simp: disjoint_iff_not_equal) text \ Two trees are {\em comparable} if they have the same alphabet and symbol frequencies. This is an important concept, because it allows us to state not only that the tree constructed by Huffman's algorithm is optimal but also that it has the expected alphabet and frequencies. We close this section with a more technical lemma. \ lemma height\<^sub>F_0_imp_Leaf_freq\<^sub>F_in_set: "\consistent\<^sub>F ts; height\<^sub>F ts = 0; a \ alphabet\<^sub>F ts\ \ Leaf (freq\<^sub>F ts a) a \ set ts" proof (induct ts) case Nil thus ?case by simp next case (Cons t ts) show ?case using Cons by (cases t) auto qed subsection \Weight\ text \ The @{term weight} function returns the weight of a tree. In the @{const Node} case, we ignore the weight cached in the node and instead compute the tree's weight recursively. This makes reasoning simpler because we can then avoid specifying cache correctness as an assumption in our lemmas. \ primrec weight :: "'a tree \ nat" where "weight (Leaf w a) = w" | "weight (Node w t\<^sub>1 t\<^sub>2) = weight t\<^sub>1 + weight t\<^sub>2" text \ The weight of a tree is the sum of the frequencies of its symbols. \myskip \noindent \isacommand{lemma} \weight_eq_Sum_freq\: \\ {\isachardoublequoteopen}$\displaystyle \consistent t \ weight t\ = \!\!\sum_{a\in @{term "alphabet t"}}^{\phantom{.}}\!\! @{term "freq t a"}$% {\isachardoublequoteclose} \vskip-\myskipamount \ (*<*) lemma weight_eq_Sum_freq: "consistent t \ weight t = (\a \ alphabet t. freq t a)" (*>*) by (induct t) (auto simp: sum.union_disjoint) text \ The assumption @{term "consistent t"} is not necessary, but it simplifies the proof by letting us invoke the lemma @{thm [source] sum.union_disjoint}: $$\\finite A; finite B; A \ B = {}\ \\~\!\sum_{x\in A} @{term "g x"} \vthinspace \mathrel{+} \sum_{x\in B} @{term "g x"}\vthinspace = % \!\!\sum_{x\in A \cup B}\! @{term "g x"}.$$ \ subsection \Cost\ text \ The {\sl cost\/} of a consistent tree, sometimes called the {\sl weighted path length}, is given by the sum $\sum_{a \in @{term "alphabet t"}\,} @{term "freq t a"} \mathbin{\*\} @{term "depth t a"}$ (which we denoted by $\sum_a w_a \vthinspace\delta_a$ in Section~\ref{binary-codes}). It obeys a simple recursive law. \ primrec cost :: "'a tree \ nat" where "cost (Leaf w a) = 0" | "cost (Node w t\<^sub>1 t\<^sub>2) = weight t\<^sub>1 + cost t\<^sub>1 + weight t\<^sub>2 + cost t\<^sub>2" text \ One interpretation of this recursive law is that the cost of a tree is the sum -of the weights of its inner nodes \<^cite>\\p.~405\ in "knuth-1997"\. (Recall that +of the weights of its inner nodes \cite[p.~405]{knuth-1997}. (Recall that $@{term "weight (Node w t\<^sub>1 t\<^sub>2)"} = @{term "weight t\<^sub>1 + weight t\<^sub>2"}$.) Since the cost of a tree is such a fundamental concept, it seems necessary to prove that the above function definition is correct. \myskip \noindent \isacommand{theorem} \cost_eq_Sum_freq_mult_depth\: \\ {\isachardoublequoteopen}$\displaystyle \consistent t \ cost t\ = \!\!\sum_{a\in @{term "alphabet t"}}^{\phantom{.}}\!\! @{term "freq t a * depth t a"}$% {\isachardoublequoteclose} \myskip \noindent The proof is by structural induction on $t$. If $t = @{term "Leaf w b"}$, both sides of the equation simplify to 0. This leaves the case $@{term t} = @{term "Node w t\<^sub>1 t\<^sub>2"}$. Let $A$, $A_1$, and $A_2$ stand for @{term "alphabet t"}, @{term "alphabet t\<^sub>1"}, and @{term "alphabet t\<^sub>2"}, respectively. We have % $$\begin{tabularx}{\textwidth}{@% {\hskip\leftmargin}cX@% {}} & @{term "cost t"} \\ \eq & \justif{definition of @{const cost}} \\ & $@{term "weight t\<^sub>1 + cost t\<^sub>1 + weight t\<^sub>2 + cost t\<^sub>2"}$ \\ \eq & \justif{induction hypothesis} \\ & $@{term "weight t\<^sub>1"} \mathrel{+} \sum_{a\in A_1\,} @{term "freq t\<^sub>1 a * depth t\<^sub>1 a"} \mathrel{+} {}$ \\ & $@{term "weight t\<^sub>2"} \mathrel{+} \sum_{a\in A_2\,} @{term "freq t\<^sub>2 a * depth t\<^sub>2 a"}$ \\ \eq & \justif{definition of @{const depth}, consistency} \\[\extrah] & $@{term "weight t\<^sub>1"} \mathrel{+} \sum_{a\in A_1\,} @{term "freq t\<^sub>1 a * (depth t a - 1)"} \mathrel{+} {}$ \\ & $@{term "weight t\<^sub>2"} \mathrel{+} \sum_{a\in A_2\,} @{term "freq t\<^sub>2 a * (depth t a - 1)"}$ \\[\extrah] \eq & \justif{distributivity of \*\ and $\sum$ over $-$} \\[\extrah] & $@{term "weight t\<^sub>1"} \mathrel{+} \sum_{a\in A_1\,} @{term "freq t\<^sub>1 a * depth t a"} \mathrel{-} \sum_{a\in A_1\,} @{term "freq t\<^sub>1 a"} \mathrel{+} {}$ \\ & $@{term "weight t\<^sub>2"} \mathrel{+} \sum_{a\in A_2\,} @{term "freq t\<^sub>2 a * depth t a"} \mathrel{-} \sum_{a\in A_2\,} @{term "freq t\<^sub>2 a"}$ \\[\extrah] \eq & \justif{@{thm [source] weight_eq_Sum_freq}} \\[\extrah] & $\sum_{a\in A_1\,} @{term "freq t\<^sub>1 a * depth t a"} \mathrel{+} \sum_{a\in A_2\,} @{term "freq t\<^sub>2 a * depth t a"}$ \\[\extrah] \eq & \justif{definition of @{const freq}, consistency} \\[\extrah] & $\sum_{a\in A_1\,} @{term "freq t a * depth t a"} \mathrel{+} \sum_{a\in A_2\,} @{term "freq t a * depth t a"}$ \\[\extrah] \eq & \justif{@{thm [source] sum.union_disjoint}, consistency} \\ & $\sum_{a\in A_1\cup A_2\,} @{term "freq t a * depth t a"}$ \\ \eq & \justif{definition of @{const alphabet}} \\ & $\sum_{a\in A\,} @{term "freq t a * depth t a"}$. \end{tabularx}$$ \noindent The structured proof closely follows this argument. \ (*<*) theorem cost_eq_Sum_freq_mult_depth: "consistent t \ cost t = (\a \ alphabet t. freq t a * depth t a)" (*>*) proof (induct t) case Leaf thus ?case by simp next case (Node w t\<^sub>1 t\<^sub>2) let ?t = "Node w t\<^sub>1 t\<^sub>2" let ?A = "alphabet ?t" and ?A\<^sub>1 = "alphabet t\<^sub>1" and ?A\<^sub>2 = "alphabet t\<^sub>2" note c = \consistent ?t\ note hyps = Node have d\<^sub>2: "\a. \?A\<^sub>1 \ ?A\<^sub>2 = {}; a \ ?A\<^sub>2\ \ depth ?t a = depth t\<^sub>2 a + 1" by auto have "cost ?t = weight t\<^sub>1 + cost t\<^sub>1 + weight t\<^sub>2 + cost t\<^sub>2" by simp also have "\ = weight t\<^sub>1 + (\a \ ?A\<^sub>1. freq t\<^sub>1 a * depth t\<^sub>1 a) + weight t\<^sub>2 + (\a \ ?A\<^sub>2. freq t\<^sub>2 a * depth t\<^sub>2 a)" using hyps by simp also have "\ = weight t\<^sub>1 + (\a \ ?A\<^sub>1. freq t\<^sub>1 a * (depth ?t a - 1)) + weight t\<^sub>2 + (\a \ ?A\<^sub>2. freq t\<^sub>2 a * (depth ?t a - 1))" using c d\<^sub>2 by simp also have "\ = weight t\<^sub>1 + (\a \ ?A\<^sub>1. freq t\<^sub>1 a * depth ?t a) - (\a \ ?A\<^sub>1. freq t\<^sub>1 a) + weight t\<^sub>2 + (\a \ ?A\<^sub>2. freq t\<^sub>2 a * depth ?t a) - (\a \ ?A\<^sub>2. freq t\<^sub>2 a)" using c d\<^sub>2 by (simp add: sum.distrib) also have "\ = (\a \ ?A\<^sub>1. freq t\<^sub>1 a * depth ?t a) + (\a \ ?A\<^sub>2. freq t\<^sub>2 a * depth ?t a)" using c by (simp add: weight_eq_Sum_freq) also have "\ = (\a \ ?A\<^sub>1. freq ?t a * depth ?t a) + (\a \ ?A\<^sub>2. freq ?t a * depth ?t a)" using c by auto also have "\ = (\a \ ?A\<^sub>1 \ ?A\<^sub>2. freq ?t a * depth ?t a)" using c by (simp add: sum.union_disjoint) also have "\ = (\a \ ?A. freq ?t a * depth ?t a)" by simp finally show ?case . qed text \ Finally, it should come as no surprise that trees with height 0 have cost 0. \ lemma height_0_imp_cost_0[simp]: "height t = 0 \ cost t = 0" by (case_tac t) simp+ subsection \Optimality\ text \ A tree is optimum if and only if its cost is not greater than that of any comparable tree. We can ignore inconsistent trees without loss of generality. \ definition optimum :: "'a tree \ bool" where "optimum t = (\u. consistent u \ alphabet t = alphabet u \ freq t = freq u \ cost t \ cost u)" section \Functional Implementation of Huffman's Algorithm \label{implementation}\ subsection \Cached Weight\ text \ The {\sl cached weight\/} of a node is the weight stored directly in the node. Our arguments rely on the computed weight (embodied by the @{const weight} function) rather than the cached weight, but the implementation of Huffman's algorithm uses the cached weight for performance reasons. \ primrec cachedWeight :: "'a tree \ nat" where "cachedWeight (Leaf w a) = w" | "cachedWeight (Node w t\<^sub>1 t\<^sub>2) = w" text \ The cached weight of a leaf node is identical to its computed weight. \ lemma height_0_imp_cachedWeight_eq_weight[simp]: "height t = 0 \ cachedWeight t = weight t" by (case_tac t) simp+ subsection \Tree Union\ text \ The implementation of Huffman's algorithm builds on two additional auxiliary functions. The first one, \uniteTrees\, takes two trees $$\vcenter{\hbox{\includegraphics[scale=1.25]{tree-w1.pdf}}} \qquad \hbox{and} \qquad \vcenter{\hbox{\includegraphics[scale=1.25]{tree-w2.pdf}}}$$ \noindent and returns the tree\strut $$\includegraphics[scale=1.25]{tree-w1-w2.pdf}$$ \vskip-.5\myskipamount \ definition uniteTrees :: "'a tree \ 'a tree \ 'a tree" where "uniteTrees t\<^sub>1 t\<^sub>2 = Node (cachedWeight t\<^sub>1 + cachedWeight t\<^sub>2) t\<^sub>1 t\<^sub>2" text \ The alphabet, consistency, and symbol frequencies of a united tree are easy to connect to the homologous properties of the subtrees. \ lemma alphabet_uniteTrees[simp]: "alphabet (uniteTrees t\<^sub>1 t\<^sub>2) = alphabet t\<^sub>1 \ alphabet t\<^sub>2" by (simp add: uniteTrees_def) lemma consistent_uniteTrees[simp]: "\consistent t\<^sub>1; consistent t\<^sub>2; alphabet t\<^sub>1 \ alphabet t\<^sub>2 = {}\ \ consistent (uniteTrees t\<^sub>1 t\<^sub>2)" by (simp add: uniteTrees_def) lemma freq_uniteTrees[simp]: "freq (uniteTrees t\<^sub>1 t\<^sub>2) a = freq t\<^sub>1 a + freq t\<^sub>2 a" by (simp add: uniteTrees_def) subsection \Ordered Tree Insertion\ text \ The auxiliary function \insortTree\ inserts a tree into a forest sorted by cached weight, preserving the sort order. \ primrec insortTree :: "'a tree \ 'a forest \ 'a forest" where "insortTree u [] = [u]" | "insortTree u (t # ts) = (if cachedWeight u \ cachedWeight t then u # t # ts else t # insortTree u ts)" text \ The resulting forest contains one more tree than the original forest. Clearly, it cannot be empty. \ lemma length_insortTree[simp]: "length (insortTree t ts) = length ts + 1" by (induct ts) simp+ lemma insortTree_ne_Nil[simp]: "insortTree t ts \ []" by (case_tac ts) simp+ text \ The alphabet, consistency, symbol frequencies, and height of a forest after insertion are easy to relate to the homologous properties of the original forest and the inserted tree. \ lemma alphabet\<^sub>F_insortTree[simp]: "alphabet\<^sub>F (insortTree t ts) = alphabet t \ alphabet\<^sub>F ts" by (induct ts) auto lemma consistent\<^sub>F_insortTree[simp]: "consistent\<^sub>F (insortTree t ts) = consistent\<^sub>F (t # ts)" by (induct ts) auto lemma freq\<^sub>F_insortTree[simp]: "freq\<^sub>F (insortTree t ts) = (\a. freq t a + freq\<^sub>F ts a)" by (induct ts) (simp add: ext)+ lemma height\<^sub>F_insortTree[simp]: "height\<^sub>F (insortTree t ts) = max (height t) (height\<^sub>F ts)" by (induct ts) auto subsection \The Main Algorithm\ text \ Huffman's algorithm repeatedly unites the first two trees of the forest it receives as argument until a single tree is left. It should initially be invoked with a list of leaf nodes sorted by weight. Note that it is not defined for the empty list. \ fun huffman :: "'a forest \ 'a tree" where "huffman [t] = t" | "huffman (t\<^sub>1 # t\<^sub>2 # ts) = huffman (insortTree (uniteTrees t\<^sub>1 t\<^sub>2) ts)" text \ The time complexity of the algorithm is quadratic in the size of the forest. If we eliminated the inner node's cached weight component, and instead recomputed the weight each time it is needed, the complexity would remain quadratic, but with a larger constant. Using a binary search in @{const insortTree}, the corresponding imperative algorithm is $O(n \log n)$ if we keep the weight cache and $O(n^2)$ if we drop it. An $O(n)$ imperative implementation is possible by maintaining two queues, one containing the unprocessed leaf nodes -and the other containing the combined trees \<^cite>\\p.~404\ in "knuth-1997"\. +and the other containing the combined trees \cite[p.~404]{knuth-1997}. The tree returned by the algorithm preserves the alphabet, consistency, and symbol frequencies of the original forest. \ theorem alphabet_huffman[simp]: "ts \ [] \ alphabet (huffman ts) = alphabet\<^sub>F ts" by (induct ts rule: huffman.induct) auto theorem consistent_huffman[simp]: "\consistent\<^sub>F ts; ts \ []\ \ consistent (huffman ts)" by (induct ts rule: huffman.induct) simp+ theorem freq_huffman[simp]: "ts \ [] \ freq (huffman ts) a = freq\<^sub>F ts a" by (induct ts rule: huffman.induct) auto section \Definition of Auxiliary Functions Used in the Proof \label{auxiliary}\ subsection \Sibling of a Symbol\ text \ The {\sl sibling\/} of a symbol $a$ in a tree $t$ is the label of the node that is the (left or right) sibling of the node labeled with $a$ in $t$. If the symbol $a$ is not in $t$'s alphabet or it occurs in a node with no sibling leaf, we define the sibling as being $a$ itself; this gives us the nice property that if $t$ is consistent, then $@{term "sibling t a"} \not= a$ if and only if $a$ has a sibling. As an illustration, we have $@{term "sibling t a"} = b$,\vthinspace{} $@{term "sibling t b"} = a$, and $@{term "sibling t c"} = c$ for the tree\strut $$t \,= \vcenter{\hbox{\includegraphics[scale=1.25]{tree-sibling.pdf}}}$$ \ fun sibling :: "'a tree \ 'a \ 'a" where "sibling (Leaf w\<^sub>b b) a = a" | "sibling (Node w (Leaf w\<^sub>b b) (Leaf w\<^sub>c c)) a = (if a = b then c else if a = c then b else a)" | "sibling (Node w t\<^sub>1 t\<^sub>2) a = (if a \ alphabet t\<^sub>1 then sibling t\<^sub>1 a else if a \ alphabet t\<^sub>2 then sibling t\<^sub>2 a else a)" text \ Because @{const sibling} is defined using sequential pattern matching -\<^cite>\"krauss-2007" and "krauss-2009"\, reasoning about it can become tedious. +\cite{krauss-2007,krauss-2009}, reasoning about it can become tedious. Simplification rules therefore play an important role. \ lemma notin_alphabet_imp_sibling_id[simp]: "a \ alphabet t \ sibling t a = a" by (cases rule: sibling.cases[where x = "(t, a)"]) simp+ lemma height_0_imp_sibling_id[simp]: "height t = 0 \ sibling t a = a" by (case_tac t) simp+ lemma height_gt_0_in_alphabet_imp_sibling_left[simp]: "\height t\<^sub>1 > 0; a \ alphabet t\<^sub>1\ \ sibling (Node w t\<^sub>1 t\<^sub>2) a = sibling t\<^sub>1 a" by (case_tac t\<^sub>1) simp+ lemma height_gt_0_in_alphabet_imp_sibling_right[simp]: "\height t\<^sub>2 > 0; a \ alphabet t\<^sub>1\ \ sibling (Node w t\<^sub>1 t\<^sub>2) a = sibling t\<^sub>1 a" by (case_tac t\<^sub>2) simp+ lemma height_gt_0_notin_alphabet_imp_sibling_left[simp]: "\height t\<^sub>1 > 0; a \ alphabet t\<^sub>1\ \ sibling (Node w t\<^sub>1 t\<^sub>2) a = sibling t\<^sub>2 a" by (case_tac t\<^sub>1) simp+ lemma height_gt_0_notin_alphabet_imp_sibling_right[simp]: "\height t\<^sub>2 > 0; a \ alphabet t\<^sub>1\ \ sibling (Node w t\<^sub>1 t\<^sub>2) a = sibling t\<^sub>2 a" by (case_tac t\<^sub>2) simp+ lemma either_height_gt_0_imp_sibling[simp]: "height t\<^sub>1 > 0 \ height t\<^sub>2 > 0 \ sibling (Node w t\<^sub>1 t\<^sub>2) a = (if a \ alphabet t\<^sub>1 then sibling t\<^sub>1 a else sibling t\<^sub>2 a)" by auto text \ The following rules are also useful for reasoning about siblings and alphabets. \ lemma in_alphabet_imp_sibling_in_alphabet: "a \ alphabet t \ sibling t a \ alphabet t" by (induct t a rule: sibling.induct) auto lemma sibling_ne_imp_sibling_in_alphabet: "sibling t a \ a \ sibling t a \ alphabet t" by (metis notin_alphabet_imp_sibling_id in_alphabet_imp_sibling_in_alphabet) text \ The default induction rule for @{const sibling} distinguishes four cases. \begin{myitemize} \item[] {\sc Base case:}\enskip $t = @{term "Leaf w b"}$. \item[] {\sc Induction step 1:}\enskip $t = @{term "Node w (Leaf w\<^sub>b b) (Leaf w\<^sub>c c)"}$. \item[] {\sc Induction step 2:}\enskip $t = @{term "Node w (Node w\<^sub>1 t\<^sub>1\<^sub>1 t\<^sub>1\<^sub>2) t\<^sub>2"}$. \item[] {\sc Induction step 3:}\enskip $t = @{term "Node w t\<^sub>1 (Node w\<^sub>2 t\<^sub>2\<^sub>1 t\<^sub>2\<^sub>2)"}$. \end{myitemize} \noindent This rule leaves much to be desired. First, the last two cases overlap and can normally be handled the same way, so they should be combined. Second, the nested \Node\ constructors in the last two cases reduce readability. Third, under the assumption that $t$ is consistent, we would like to perform the same case distinction on $a$ as we did for @{thm [source] tree_induct_consistent}, with the same benefits for automation. These observations lead us to develop a custom induction rule that distinguishes the following cases. \begin{myitemize} \item[] {\sc Base case:}\enskip $t = @{term "Leaf w b"}$. \item[] {\sc Induction step 1:}\enskip $t = @{term "Node w (Leaf w\<^sub>b b) (Leaf w\<^sub>c c)"}$ with @{prop "b \ c"}. \item[] \begin{flushleft} {\sc Induction step 2:}\enskip $t = @{term "Node w t\<^sub>1 t\<^sub>2"}$ and either @{term t\<^sub>1} or @{term t\<^sub>2} has nonzero height. \end{flushleft} \item[] \noindent\kern\leftmargin {\sc Subcase 1:}\enspace $a$ belongs to @{term t\<^sub>1} but not to @{term t\<^sub>2}. \item[] \noindent\kern\leftmargin {\sc Subcase 2:}\enspace $a$ belongs to @{term t\<^sub>2} but not to @{term t\<^sub>1}. \item[] \noindent\kern\leftmargin {\sc Subcase 3:}\enspace $a$ belongs to neither @{term t\<^sub>1} nor @{term t\<^sub>2}. \end{myitemize} The statement of the rule and its proof are similar to what we did for consistent trees, the main difference being that we now have two induction steps instead of one. \ lemma sibling_induct_consistent[consumes 1, case_names base step\<^sub>1 step\<^sub>2\<^sub>1 step\<^sub>2\<^sub>2 step\<^sub>2\<^sub>3]: "\consistent t; \w b a. P (Leaf w b) a; \w w\<^sub>b b w\<^sub>c c a. b \ c \ P (Node w (Leaf w\<^sub>b b) (Leaf w\<^sub>c c)) a; \w t\<^sub>1 t\<^sub>2 a. \consistent t\<^sub>1; consistent t\<^sub>2; alphabet t\<^sub>1 \ alphabet t\<^sub>2 = {}; height t\<^sub>1 > 0 \ height t\<^sub>2 > 0; a \ alphabet t\<^sub>1; sibling t\<^sub>1 a \ alphabet t\<^sub>1; a \ alphabet t\<^sub>2; sibling t\<^sub>1 a \ alphabet t\<^sub>2; P t\<^sub>1 a\ \ P (Node w t\<^sub>1 t\<^sub>2) a; \w t\<^sub>1 t\<^sub>2 a. \consistent t\<^sub>1; consistent t\<^sub>2; alphabet t\<^sub>1 \ alphabet t\<^sub>2 = {}; height t\<^sub>1 > 0 \ height t\<^sub>2 > 0; a \ alphabet t\<^sub>1; sibling t\<^sub>2 a \ alphabet t\<^sub>1; a \ alphabet t\<^sub>2; sibling t\<^sub>2 a \ alphabet t\<^sub>2; P t\<^sub>2 a\ \ P (Node w t\<^sub>1 t\<^sub>2) a; \w t\<^sub>1 t\<^sub>2 a. \consistent t\<^sub>1; consistent t\<^sub>2; alphabet t\<^sub>1 \ alphabet t\<^sub>2 = {}; height t\<^sub>1 > 0 \ height t\<^sub>2 > 0; a \ alphabet t\<^sub>1; a \ alphabet t\<^sub>2\ \ P (Node w t\<^sub>1 t\<^sub>2) a\ \ P t a" apply rotate_tac apply induction_schema apply atomize_elim apply (case_tac t, simp) apply clarsimp apply (rename_tac a t\<^sub>1 t\<^sub>2) apply (case_tac "height t\<^sub>1 = 0 \ height t\<^sub>2 = 0") apply simp apply (case_tac t\<^sub>1) apply (case_tac t\<^sub>2) apply fastforce apply simp+ apply (auto intro: in_alphabet_imp_sibling_in_alphabet)[1] by lexicographic_order text \ The custom induction rule allows us to prove new properties of @{const sibling} with little effort. \ lemma sibling_sibling_id[simp]: "consistent t \ sibling t (sibling t a) = a" by (induct t a rule: sibling_induct_consistent) simp+ lemma sibling_reciprocal: "\consistent t; sibling t a = b\ \ sibling t b = a" by auto lemma depth_height_imp_sibling_ne: "\consistent t; depth t a = height t; height t > 0; a \ alphabet t\ \ sibling t a \ a" by (induct t a rule: sibling_induct_consistent) auto lemma depth_sibling[simp]: "consistent t \ depth t (sibling t a) = depth t a" by (induct t a rule: sibling_induct_consistent) simp+ subsection \Leaf Interchange\ text \ The \swapLeaves\ function takes a tree $t$ together with two symbols $a$, $b$ and their frequencies $@{term w\<^sub>a}$, $@{term w\<^sub>b}$, and returns the tree $t$ in which the leaf nodes labeled with $a$ and $b$ are exchanged. When invoking \swapLeaves\, we normally pass @{term "freq t a"} and @{term "freq t b"} for @{term w\<^sub>a} and @{term w\<^sub>b}. Note that we do not bother updating the cached weight of the ancestor nodes when performing the interchange. The cached weight is used only in the implementation of Huffman's algorithm, which does not invoke \swapLeaves\. \ primrec swapLeaves :: "'a tree \ nat \ 'a \ nat \ 'a \ 'a tree" where "swapLeaves (Leaf w\<^sub>c c) w\<^sub>a a w\<^sub>b b = (if c = a then Leaf w\<^sub>b b else if c = b then Leaf w\<^sub>a a else Leaf w\<^sub>c c)" | "swapLeaves (Node w t\<^sub>1 t\<^sub>2) w\<^sub>a a w\<^sub>b b = Node w (swapLeaves t\<^sub>1 w\<^sub>a a w\<^sub>b b) (swapLeaves t\<^sub>2 w\<^sub>a a w\<^sub>b b)" text \ Swapping a symbol~$a$ with itself leaves the tree $t$ unchanged if $a$ does not belong to it or if the specified frequencies @{term w\<^sub>a} and @{term w\<^sub>b} equal @{term "freq t a"}. \ lemma swapLeaves_id_when_notin_alphabet[simp]: "a \ alphabet t \ swapLeaves t w a w' a = t" by (induct t) simp+ lemma swapLeaves_id[simp]: "consistent t \ swapLeaves t (freq t a) a (freq t a) a = t" by (induct t a rule: tree_induct_consistent) simp+ text \ The alphabet, consistency, symbol depths, height, and symbol frequencies of the tree @{term "swapLeaves t w\<^sub>a a w\<^sub>b b"} can be related to the homologous properties of $t$. \ lemma alphabet_swapLeaves: "alphabet (swapLeaves t w\<^sub>a a w\<^sub>b b) = (if a \ alphabet t then if b \ alphabet t then alphabet t else (alphabet t - {a}) \ {b} else if b \ alphabet t then (alphabet t - {b}) \ {a} else alphabet t)" by (induct t) auto lemma consistent_swapLeaves[simp]: "consistent t \ consistent (swapLeaves t w\<^sub>a a w\<^sub>b b)" by (induct t) (auto simp: alphabet_swapLeaves) lemma depth_swapLeaves_neither[simp]: "\consistent t; c \ a; c \ b\ \ depth (swapLeaves t w\<^sub>a a w\<^sub>b b) c = depth t c" by (induct t a rule: tree_induct_consistent) (auto simp: alphabet_swapLeaves) lemma height_swapLeaves[simp]: "height (swapLeaves t w\<^sub>a a w\<^sub>b b) = height t" by (induct t) simp+ lemma freq_swapLeaves[simp]: "\consistent t; a \ b\ \ freq (swapLeaves t w\<^sub>a a w\<^sub>b b) = (\c. if c = a then if b \ alphabet t then w\<^sub>a else 0 else if c = b then if a \ alphabet t then w\<^sub>b else 0 else freq t c)" apply (rule ext) apply (induct t) by auto text \ For the lemmas concerned with the resulting tree's weight and cost, we avoid subtraction on natural numbers by rearranging terms. For example, we write $$@{prop "weight (swapLeaves t w\<^sub>a a w\<^sub>b b) + freq t a = weight t + w\<^sub>b"}$$ \noindent rather than the more conventional $$@{prop "weight (swapLeaves t w\<^sub>a a w\<^sub>b b) = weight t + w\<^sub>b - freq t a"}.$$ In Isabelle/HOL, these two equations are not equivalent, because by definition $m - n = 0$ if $n > m$. We could use the second equation and additionally assert that @{prop "weight t \ freq t a"} (an easy consequence of @{thm [source] weight_eq_Sum_freq}), and then apply the \textit{arith} tactic, but it is much simpler to use the first equation and stay with \textit{simp} and \textit{auto}. Another option would be to use integers instead of natural numbers. \ lemma weight_swapLeaves: "\consistent t; a \ b\ \ if a \ alphabet t then if b \ alphabet t then weight (swapLeaves t w\<^sub>a a w\<^sub>b b) + freq t a + freq t b = weight t + w\<^sub>a + w\<^sub>b else weight (swapLeaves t w\<^sub>a a w\<^sub>b b) + freq t a = weight t + w\<^sub>b else if b \ alphabet t then weight (swapLeaves t w\<^sub>a a w\<^sub>b b) + freq t b = weight t + w\<^sub>a else weight (swapLeaves t w\<^sub>a a w\<^sub>b b) = weight t" proof (induct t a rule: tree_induct_consistent) \ \{\sc Base case:}\enspace $t = @{term "Leaf w b"}$\ case base thus ?case by clarsimp next \ \{\sc Induction step:}\enspace $t = @{term "Node w t\<^sub>1 t\<^sub>2"}$\ \ \{\sc Subcase 1:}\enspace $a$ belongs to @{term t\<^sub>1} but not to @{term t\<^sub>2}\ case (step\<^sub>1 w t\<^sub>1 t\<^sub>2 a) show ?case proof cases assume b: "b \ alphabet t\<^sub>1" hence "b \ alphabet t\<^sub>2" using step\<^sub>1 by auto thus ?case using b step\<^sub>1 by simp next assume "b \ alphabet t\<^sub>1" thus ?case using step\<^sub>1 by auto qed next \ \{\sc Subcase 2:}\enspace $a$ belongs to @{term t\<^sub>2} but not to @{term t\<^sub>1}\ case (step\<^sub>2 w t\<^sub>1 t\<^sub>2 a) show ?case proof cases assume b: "b \ alphabet t\<^sub>1" hence "b \ alphabet t\<^sub>2" using step\<^sub>2 by auto thus ?case using b step\<^sub>2 by simp next assume "b \ alphabet t\<^sub>1" thus ?case using step\<^sub>2 by auto qed next \ \{\sc Subcase 3:}\enspace $a$ belongs to neither @{term t\<^sub>1} nor @{term t\<^sub>2}\ case (step\<^sub>3 w t\<^sub>1 t\<^sub>2 a) show ?case proof cases assume b: "b \ alphabet t\<^sub>1" hence "b \ alphabet t\<^sub>2" using step\<^sub>3 by auto thus ?case using b step\<^sub>3 by simp next assume "b \ alphabet t\<^sub>1" thus ?case using step\<^sub>3 by auto qed qed lemma cost_swapLeaves: "\consistent t; a \ b\ \ if a \ alphabet t then if b \ alphabet t then cost (swapLeaves t w\<^sub>a a w\<^sub>b b) + freq t a * depth t a + freq t b * depth t b = cost t + w\<^sub>a * depth t b + w\<^sub>b * depth t a else cost (swapLeaves t w\<^sub>a a w\<^sub>b b) + freq t a * depth t a = cost t + w\<^sub>b * depth t a else if b \ alphabet t then cost (swapLeaves t w\<^sub>a a w\<^sub>b b) + freq t b * depth t b = cost t + w\<^sub>a * depth t b else cost (swapLeaves t w\<^sub>a a w\<^sub>b b) = cost t" proof (induct t) case Leaf show ?case by simp next case (Node w t\<^sub>1 t\<^sub>2) note c = \consistent (Node w t\<^sub>1 t\<^sub>2)\ note hyps = Node have w\<^sub>1: "if a \ alphabet t\<^sub>1 then if b \ alphabet t\<^sub>1 then weight (swapLeaves t\<^sub>1 w\<^sub>a a w\<^sub>b b) + freq t\<^sub>1 a + freq t\<^sub>1 b = weight t\<^sub>1 + w\<^sub>a + w\<^sub>b else weight (swapLeaves t\<^sub>1 w\<^sub>a a w\<^sub>b b) + freq t\<^sub>1 a = weight t\<^sub>1 + w\<^sub>b else if b \ alphabet t\<^sub>1 then weight (swapLeaves t\<^sub>1 w\<^sub>a a w\<^sub>b b) + freq t\<^sub>1 b = weight t\<^sub>1 + w\<^sub>a else weight (swapLeaves t\<^sub>1 w\<^sub>a a w\<^sub>b b) = weight t\<^sub>1" using hyps by (simp add: weight_swapLeaves) have w\<^sub>2: "if a \ alphabet t\<^sub>2 then if b \ alphabet t\<^sub>2 then weight (swapLeaves t\<^sub>2 w\<^sub>a a w\<^sub>b b) + freq t\<^sub>2 a + freq t\<^sub>2 b = weight t\<^sub>2 + w\<^sub>a + w\<^sub>b else weight (swapLeaves t\<^sub>2 w\<^sub>a a w\<^sub>b b) + freq t\<^sub>2 a = weight t\<^sub>2 + w\<^sub>b else if b \ alphabet t\<^sub>2 then weight (swapLeaves t\<^sub>2 w\<^sub>a a w\<^sub>b b) + freq t\<^sub>2 b = weight t\<^sub>2 + w\<^sub>a else weight (swapLeaves t\<^sub>2 w\<^sub>a a w\<^sub>b b) = weight t\<^sub>2" using hyps by (simp add: weight_swapLeaves) show ?case proof cases assume a\<^sub>1: "a \ alphabet t\<^sub>1" hence a\<^sub>2: "a \ alphabet t\<^sub>2" using c by auto show ?case proof cases assume b\<^sub>1: "b \ alphabet t\<^sub>1" hence "b \ alphabet t\<^sub>2" using c by auto thus ?case using a\<^sub>1 a\<^sub>2 b\<^sub>1 w\<^sub>1 w\<^sub>2 hyps by simp next assume b\<^sub>1: "b \ alphabet t\<^sub>1" show ?case proof cases assume "b \ alphabet t\<^sub>2" thus ?case using a\<^sub>1 a\<^sub>2 b\<^sub>1 w\<^sub>1 w\<^sub>2 hyps by simp next assume "b \ alphabet t\<^sub>2" thus ?case using a\<^sub>1 a\<^sub>2 b\<^sub>1 w\<^sub>1 w\<^sub>2 hyps by simp qed qed next assume a\<^sub>1: "a \ alphabet t\<^sub>1" show ?case proof cases assume a\<^sub>2: "a \ alphabet t\<^sub>2" show ?case proof cases assume b\<^sub>1: "b \ alphabet t\<^sub>1" hence "b \ alphabet t\<^sub>2" using c by auto thus ?case using a\<^sub>1 a\<^sub>2 b\<^sub>1 w\<^sub>1 w\<^sub>2 hyps by simp next assume b\<^sub>1: "b \ alphabet t\<^sub>1" show ?case proof cases assume "b \ alphabet t\<^sub>2" thus ?case using a\<^sub>1 a\<^sub>2 b\<^sub>1 w\<^sub>1 w\<^sub>2 hyps by simp next assume "b \ alphabet t\<^sub>2" thus ?case using a\<^sub>1 a\<^sub>2 b\<^sub>1 w\<^sub>1 w\<^sub>2 hyps by simp qed qed next assume a\<^sub>2: "a \ alphabet t\<^sub>2" show ?case proof cases assume b\<^sub>1: "b \ alphabet t\<^sub>1" hence "b \ alphabet t\<^sub>2" using c by auto thus ?case using a\<^sub>1 a\<^sub>2 b\<^sub>1 w\<^sub>1 w\<^sub>2 hyps by simp next assume b\<^sub>1: "b \ alphabet t\<^sub>1" show ?case proof cases assume "b \ alphabet t\<^sub>2" thus ?case using a\<^sub>1 a\<^sub>2 b\<^sub>1 w\<^sub>1 w\<^sub>2 hyps by simp next assume "b \ alphabet t\<^sub>2" thus ?case using a\<^sub>1 a\<^sub>2 b\<^sub>1 w\<^sub>1 w\<^sub>2 hyps by simp qed qed qed qed qed text \ Common sense tells us that the following statement is valid: ``If Astrid exchanges her house with Bernard's neighbor, Bernard becomes Astrid's new neighbor.'' A similar property holds for binary trees. \ lemma sibling_swapLeaves_sibling[simp]: "\consistent t; sibling t b \ b; a \ b\ \ sibling (swapLeaves t w\<^sub>a a w\<^sub>s (sibling t b)) a = b" proof (induct t) case Leaf thus ?case by simp next case (Node w t\<^sub>1 t\<^sub>2) note hyps = Node show ?case proof (cases "height t\<^sub>1 = 0") case True note h\<^sub>1 = True show ?thesis proof (cases t\<^sub>1) case (Leaf w\<^sub>c c) note l\<^sub>1 = Leaf show ?thesis proof (cases "height t\<^sub>2 = 0") case True note h\<^sub>2 = True show ?thesis proof (cases t\<^sub>2) case Leaf thus ?thesis using l\<^sub>1 hyps by auto metis+ next case Node thus ?thesis using h\<^sub>2 by simp qed next case False note h\<^sub>2 = False show ?thesis proof cases assume "c = b" thus ?thesis using l\<^sub>1 h\<^sub>2 hyps by simp next assume "c \ b" have "sibling t\<^sub>2 b \ alphabet t\<^sub>2" using \c \ b\ l\<^sub>1 h\<^sub>2 hyps by (simp add: sibling_ne_imp_sibling_in_alphabet) thus ?thesis using \c \ b\ l\<^sub>1 h\<^sub>2 hyps by auto qed qed next case Node thus ?thesis using h\<^sub>1 by simp qed next case False note h\<^sub>1 = False show ?thesis proof (cases "height t\<^sub>2 = 0") case True note h\<^sub>2 = True show ?thesis proof (cases t\<^sub>2) case (Leaf w\<^sub>d d) note l\<^sub>2 = Leaf show ?thesis proof cases assume "d = b" thus ?thesis using h\<^sub>1 l\<^sub>2 hyps by simp next assume "d \ b" show ?thesis proof (cases "b \ alphabet t\<^sub>1") case True hence "sibling t\<^sub>1 b \ alphabet t\<^sub>1" using \d \ b\ h\<^sub>1 l\<^sub>2 hyps by (simp add: sibling_ne_imp_sibling_in_alphabet) thus ?thesis using True \d \ b\ h\<^sub>1 l\<^sub>2 hyps by (simp add: alphabet_swapLeaves) next case False thus ?thesis using \d \ b\ l\<^sub>2 hyps by simp qed qed next case Node thus ?thesis using h\<^sub>2 by simp qed next case False note h\<^sub>2 = False show ?thesis proof (cases "b \ alphabet t\<^sub>1") case True thus ?thesis using h\<^sub>1 h\<^sub>2 hyps by auto next case False note b\<^sub>1 = False show ?thesis proof (cases "b \ alphabet t\<^sub>2") case True thus ?thesis using b\<^sub>1 h\<^sub>1 h\<^sub>2 hyps by (auto simp: in_alphabet_imp_sibling_in_alphabet alphabet_swapLeaves) next case False thus ?thesis using b\<^sub>1 h\<^sub>1 h\<^sub>2 hyps by simp qed qed qed qed qed subsection \Symbol Interchange\ text \ The \swapSyms\ function provides a simpler interface to @{const swapLeaves}, with @{term "freq t a"} and @{term "freq t b"} in place of @{term "w\<^sub>a"} and @{term "w\<^sub>b"}. Most lemmas about \swapSyms\ are directly adapted from the homologous results about @{const swapLeaves}. \ definition swapSyms :: "'a tree \ 'a \ 'a \ 'a tree" where "swapSyms t a b = swapLeaves t (freq t a) a (freq t b) b" lemma swapSyms_id[simp]: "consistent t \ swapSyms t a a = t" by (simp add: swapSyms_def) lemma alphabet_swapSyms[simp]: "\a \ alphabet t; b \ alphabet t\ \ alphabet (swapSyms t a b) = alphabet t" by (simp add: swapSyms_def alphabet_swapLeaves) lemma consistent_swapSyms[simp]: "consistent t \ consistent (swapSyms t a b)" by (simp add: swapSyms_def) lemma depth_swapSyms_neither[simp]: "\consistent t; c \ a; c \ b\ \ depth (swapSyms t a b) c = depth t c" by (simp add: swapSyms_def) lemma freq_swapSyms[simp]: "\consistent t; a \ alphabet t; b \ alphabet t\ \ freq (swapSyms t a b) = freq t" by (case_tac "a = b") (simp add: swapSyms_def ext)+ lemma cost_swapSyms: assumes "consistent t" "a \ alphabet t" "b \ alphabet t" shows "cost (swapSyms t a b) + freq t a * depth t a + freq t b * depth t b = cost t + freq t a * depth t b + freq t b * depth t a" proof cases assume "a = b" thus ?thesis using assms by simp next assume "a \ b" hence "cost (swapLeaves t (freq t a) a (freq t b) b) + freq t a * depth t a + freq t b * depth t b = cost t + freq t a * depth t b + freq t b * depth t a" using assms by (simp add: cost_swapLeaves) thus ?thesis using assms by (simp add: swapSyms_def) qed text \ If $a$'s frequency is lower than or equal to $b$'s, and $a$ is higher up in the tree than $b$ or at the same level, then interchanging $a$ and $b$ does not increase the tree's cost. \ lemma le_le_imp_sum_mult_le_sum_mult: assumes "i \ j" "m \ (n::nat)" shows "i * n + j * m \ i * m + j * n" proof - have "i * m + i * (n - m) + j * m \ i * m + j * m + j * (n - m)" using assms by simp thus ?thesis using assms by (simp add: diff_mult_distrib2) qed lemma cost_swapSyms_le: assumes "consistent t" "a \ alphabet t" "b \ alphabet t" "freq t a \ freq t b" "depth t a \ depth t b" shows "cost (swapSyms t a b) \ cost t" proof - let ?aabb = "freq t a * depth t a + freq t b * depth t b" let ?abba = "freq t a * depth t b + freq t b * depth t a" have "?abba \ ?aabb" using assms(4-5) by (rule le_le_imp_sum_mult_le_sum_mult) have "cost (swapSyms t a b) + ?aabb = cost t + ?abba" using assms(1-3) by (simp add: cost_swapSyms add.assoc[THEN sym]) also have "\ \ cost t + ?aabb" using \?abba \ ?aabb\ by simp finally show ?thesis using assms(4-5) by simp qed text \ As stated earlier, ``If Astrid exchanges her house with Bernard's neighbor, Bernard becomes Astrid's new neighbor.'' \ lemma sibling_swapSyms_sibling[simp]: "\consistent t; sibling t b \ b; a \ b\ \ sibling (swapSyms t a (sibling t b)) a = b" by (simp add: swapSyms_def) text \ ``If Astrid exchanges her house with Bernard, Astrid becomes Bernard's old neighbor's new neighbor.'' \ lemma sibling_swapSyms_other_sibling[simp]: "\consistent t; sibling t b \ a; sibling t b \ b; a \ b\ \ sibling (swapSyms t a b) (sibling t b) = a" by (metis consistent_swapSyms sibling_swapSyms_sibling sibling_reciprocal) subsection \Four-Way Symbol Interchange \label{four-way-symbol-interchange}\ text \ The @{const swapSyms} function exchanges two symbols $a$ and $b$. We use it to define the four-way symbol interchange function \swapFourSyms\, which takes four symbols $a$, $b$, $c$, $d$ with $a \ne b$ and $c \ne d$, and exchanges them so that $a$ and $b$ occupy $c$~and~$d$'s positions. A naive definition of this function would be $$@{prop "swapFourSyms t a b c d = swapSyms (swapSyms t a c) b d"}.$$ This definition fails in the face of aliasing: If $a = d$, but $b \ne c$, then \swapFourSyms a b c d\ would leave $a$ in $b$'s position.% -\footnote{Cormen et al.\ \<^cite>\\p.~390\ in "cormen-et-al-2001"\ forgot to consider +\footnote{Cormen et al.\ \cite[p.~390]{cormen-et-al-2001} forgot to consider this case in their proof. Thomas Cormen indicated in a personal communication that this will be corrected in the next edition of the book.} \ definition swapFourSyms :: "'a tree \ 'a \ 'a \ 'a \ 'a \ 'a tree" where "swapFourSyms t a b c d = (if a = d then swapSyms t b c else if b = c then swapSyms t a d else swapSyms (swapSyms t a c) b d)" text \ Lemmas about @{const swapFourSyms} are easy to prove by expanding its definition. \ lemma alphabet_swapFourSyms[simp]: "\a \ alphabet t; b \ alphabet t; c \ alphabet t; d \ alphabet t\ \ alphabet (swapFourSyms t a b c d) = alphabet t" by (simp add: swapFourSyms_def) lemma consistent_swapFourSyms[simp]: "consistent t \ consistent (swapFourSyms t a b c d)" by (simp add: swapFourSyms_def) lemma freq_swapFourSyms[simp]: "\consistent t; a \ alphabet t; b \ alphabet t; c \ alphabet t; d \ alphabet t\ \ freq (swapFourSyms t a b c d) = freq t" by (auto simp: swapFourSyms_def) text \ ``If Astrid and Bernard exchange their houses with Carmen and her neighbor, Astrid and Bernard will now be neighbors.'' \ lemma sibling_swapFourSyms_when_4th_is_sibling: assumes "consistent t" "a \ alphabet t" "b \ alphabet t" "c \ alphabet t" "a \ b" "sibling t c \ c" shows "sibling (swapFourSyms t a b c (sibling t c)) a = b" proof (cases "a \ sibling t c \ b \ c") case True show ?thesis proof - let ?d = "sibling t c" let ?t\<^sub>s = "swapFourSyms t a b c ?d" have abba: "(sibling ?t\<^sub>s a = b) = (sibling ?t\<^sub>s b = a)" using \consistent t\ by (metis consistent_swapFourSyms sibling_reciprocal) have s: "sibling t c = sibling (swapSyms t a c) a" using True assms by (metis sibling_reciprocal sibling_swapSyms_sibling) have "sibling ?t\<^sub>s b = sibling (swapSyms t a c) ?d" using s True assms by (auto simp: swapFourSyms_def) also have "\ = a" using True assms by (metis sibling_reciprocal sibling_swapSyms_other_sibling swapLeaves_id swapSyms_def) finally have "sibling ?t\<^sub>s b = a" . with abba show ?thesis .. qed next case False thus ?thesis using assms by (auto intro: sibling_reciprocal simp: swapFourSyms_def) qed subsection \Sibling Merge\ text \ Given a symbol $a$, the \mergeSibling\ function transforms the tree % \setbox\myboxi=\hbox{\includegraphics[scale=1.25]{tree-splitLeaf-a.pdf}}% \setbox\myboxii=\hbox{\includegraphics[scale=1.25]{tree-splitLeaf-ab.pdf}}% \mydimeni=\ht\myboxii $$\vcenter{\box\myboxii} \qquad \hbox{into} \qquad \smash{\lower\ht\myboxi\hbox{\raise.5\mydimeni\box\myboxi}}$$ The frequency of $a$ in the result is the sum of the original frequencies of $a$ and $b$, so as not to alter the tree's weight. \ fun mergeSibling :: "'a tree \ 'a \ 'a tree" where "mergeSibling (Leaf w\<^sub>b b) a = Leaf w\<^sub>b b" | "mergeSibling (Node w (Leaf w\<^sub>b b) (Leaf w\<^sub>c c)) a = (if a = b \ a = c then Leaf (w\<^sub>b + w\<^sub>c) a else Node w (Leaf w\<^sub>b b) (Leaf w\<^sub>c c))" | "mergeSibling (Node w t\<^sub>1 t\<^sub>2) a = Node w (mergeSibling t\<^sub>1 a) (mergeSibling t\<^sub>2 a)" text \ The definition of @{const mergeSibling} has essentially the same structure as that of @{const sibling}. As a result, the custom induction rule that we derived for @{const sibling} works equally well for reasoning about @{const mergeSibling}. \ lemmas mergeSibling_induct_consistent = sibling_induct_consistent text \ The properties of @{const mergeSibling} echo those of @{const sibling}. Like with @{const sibling}, simplification rules are crucial. \ lemma notin_alphabet_imp_mergeSibling_id[simp]: "a \ alphabet t \ mergeSibling t a = t" by (induct t a rule: mergeSibling.induct) simp+ lemma height_gt_0_imp_mergeSibling_left[simp]: "height t\<^sub>1 > 0 \ mergeSibling (Node w t\<^sub>1 t\<^sub>2) a = Node w (mergeSibling t\<^sub>1 a) (mergeSibling t\<^sub>2 a)" by (case_tac t\<^sub>1) simp+ lemma height_gt_0_imp_mergeSibling_right[simp]: "height t\<^sub>2 > 0 \ mergeSibling (Node w t\<^sub>1 t\<^sub>2) a = Node w (mergeSibling t\<^sub>1 a) (mergeSibling t\<^sub>2 a)" by (case_tac t\<^sub>2) simp+ lemma either_height_gt_0_imp_mergeSibling[simp]: "height t\<^sub>1 > 0 \ height t\<^sub>2 > 0 \ mergeSibling (Node w t\<^sub>1 t\<^sub>2) a = Node w (mergeSibling t\<^sub>1 a) (mergeSibling t\<^sub>2 a)" by auto lemma alphabet_mergeSibling[simp]: "\consistent t; a \ alphabet t\ \ alphabet (mergeSibling t a) = (alphabet t - {sibling t a}) \ {a}" by (induct t a rule: mergeSibling_induct_consistent) auto lemma consistent_mergeSibling[simp]: "consistent t \ consistent (mergeSibling t a)" by (induct t a rule: mergeSibling_induct_consistent) auto lemma freq_mergeSibling: "\consistent t; a \ alphabet t; sibling t a \ a\ \ freq (mergeSibling t a) = (\c. if c = a then freq t a + freq t (sibling t a) else if c = sibling t a then 0 else freq t c)" by (induct t a rule: mergeSibling_induct_consistent) (auto simp: fun_eq_iff) lemma weight_mergeSibling[simp]: "weight (mergeSibling t a) = weight t" by (induct t a rule: mergeSibling.induct) simp+ text \ If $a$ has a sibling, merging $a$ and its sibling reduces $t$'s cost by @{term "freq t a + freq t (sibling t a)"}. \ lemma cost_mergeSibling: "\consistent t; sibling t a \ a\ \ cost (mergeSibling t a) + freq t a + freq t (sibling t a) = cost t" by (induct t a rule: mergeSibling_induct_consistent) auto subsection \Leaf Split\ text \ The \splitLeaf\ function undoes the merging performed by @{const mergeSibling}: Given two symbols $a$, $b$ and two frequencies $@{term w\<^sub>a}$, $@{term w\<^sub>b}$, it transforms \setbox\myboxi=\hbox{\includegraphics[scale=1.25]{tree-splitLeaf-a.pdf}}% \setbox\myboxii=\hbox{\includegraphics[scale=1.25]{tree-splitLeaf-ab.pdf}}% $$\smash{\lower\ht\myboxi\hbox{\raise.5\ht\myboxii\box\myboxi}} \qquad \hbox{into} \qquad \vcenter{\box\myboxii}$$ In the resulting tree, $a$ has frequency @{term w\<^sub>a} and $b$ has frequency @{term w\<^sub>b}. We normally invoke it with @{term w\<^sub>a}~and @{term w\<^sub>b} such that @{prop "freq t a = w\<^sub>a + w\<^sub>b"}. \ primrec splitLeaf :: "'a tree \ nat \ 'a \ nat \ 'a \ 'a tree" where "splitLeaf (Leaf w\<^sub>c c) w\<^sub>a a w\<^sub>b b = (if c = a then Node w\<^sub>c (Leaf w\<^sub>a a) (Leaf w\<^sub>b b) else Leaf w\<^sub>c c)" | "splitLeaf (Node w t\<^sub>1 t\<^sub>2) w\<^sub>a a w\<^sub>b b = Node w (splitLeaf t\<^sub>1 w\<^sub>a a w\<^sub>b b) (splitLeaf t\<^sub>2 w\<^sub>a a w\<^sub>b b)" primrec splitLeaf\<^sub>F :: "'a forest \ nat \ 'a \ nat \ 'a \ 'a forest" where "splitLeaf\<^sub>F [] w\<^sub>a a w\<^sub>b b = []" | "splitLeaf\<^sub>F (t # ts) w\<^sub>a a w\<^sub>b b = splitLeaf t w\<^sub>a a w\<^sub>b b # splitLeaf\<^sub>F ts w\<^sub>a a w\<^sub>b b" text \ Splitting leaf nodes affects the alphabet, consistency, symbol frequencies, weight, and cost in unsurprising ways. \ lemma notin_alphabet_imp_splitLeaf_id[simp]: "a \ alphabet t \ splitLeaf t w\<^sub>a a w\<^sub>b b = t" by (induct t) simp+ lemma notin_alphabet\<^sub>F_imp_splitLeaf\<^sub>F_id[simp]: "a \ alphabet\<^sub>F ts \ splitLeaf\<^sub>F ts w\<^sub>a a w\<^sub>b b = ts" by (induct ts) simp+ lemma alphabet_splitLeaf[simp]: "alphabet (splitLeaf t w\<^sub>a a w\<^sub>b b) = (if a \ alphabet t then alphabet t \ {b} else alphabet t)" by (induct t) simp+ lemma consistent_splitLeaf[simp]: "\consistent t; b \ alphabet t\ \ consistent (splitLeaf t w\<^sub>a a w\<^sub>b b)" by (induct t) auto lemma freq_splitLeaf[simp]: "\consistent t; b \ alphabet t\ \ freq (splitLeaf t w\<^sub>a a w\<^sub>b b) = (if a \ alphabet t then (\c. if c = a then w\<^sub>a else if c = b then w\<^sub>b else freq t c) else freq t)" by (induct t b rule: tree_induct_consistent) (rule ext, auto)+ lemma weight_splitLeaf[simp]: "\consistent t; a \ alphabet t; freq t a = w\<^sub>a + w\<^sub>b\ \ weight (splitLeaf t w\<^sub>a a w\<^sub>b b) = weight t" by (induct t a rule: tree_induct_consistent) simp+ lemma cost_splitLeaf[simp]: "\consistent t; a \ alphabet t; freq t a = w\<^sub>a + w\<^sub>b\ \ cost (splitLeaf t w\<^sub>a a w\<^sub>b b) = cost t + w\<^sub>a + w\<^sub>b" by (induct t a rule: tree_induct_consistent) simp+ subsection \Weight Sort Order\ text \ An invariant of Huffman's algorithm is that the forest is sorted by weight. This is expressed by the \sortedByWeight\ function. \ fun sortedByWeight :: "'a forest \ bool" where "sortedByWeight [] = True" | "sortedByWeight [t] = True" | "sortedByWeight (t\<^sub>1 # t\<^sub>2 # ts) = (weight t\<^sub>1 \ weight t\<^sub>2 \ sortedByWeight (t\<^sub>2 # ts))" text \ The function obeys the following fairly obvious laws. \ lemma sortedByWeight_Cons_imp_sortedByWeight: "sortedByWeight (t # ts) \ sortedByWeight ts" by (case_tac ts) simp+ lemma sortedByWeight_Cons_imp_forall_weight_ge: "sortedByWeight (t # ts) \ \u \ set ts. weight u \ weight t" by (induct ts arbitrary: t) force+ lemma sortedByWeight_insortTree: "\sortedByWeight ts; height t = 0; height\<^sub>F ts = 0\ \ sortedByWeight (insortTree t ts)" by (induct ts rule: sortedByWeight.induct) auto subsection \Pair of Minimal Symbols\ text \ The \minima\ predicate expresses that two symbols $a$, $b \in @{term "alphabet t"}$ have the lowest frequencies in the tree $t$. Minimal symbols need not be uniquely defined. \ definition minima :: "'a tree \ 'a \ 'a \ bool" where "minima t a b = (a \ alphabet t \ b \ alphabet t \ a \ b \ (\c \ alphabet t. c \ a \ c \ b \ freq t c \ freq t a \ freq t c \ freq t b))" section \Formalization of the Textbook Proof \label{formalization}\ subsection \Four-Way Symbol Interchange Cost Lemma\ text \ If $a$ and $b$ are minima, and $c$ and $d$ are at the very bottom of the tree, then exchanging $a$ and $b$ with $c$ and $d$ does not increase the cost. Graphically, we have\strut % $${\it cost\/} \vcenter{\hbox{\includegraphics[scale=1.25]{tree-minima-abcd.pdf}}} \,\mathop{\le}\;\;\; {\it cost\/} \vcenter{\hbox{\includegraphics[scale=1.25]{tree-minima.pdf}}}$$ \noindent This cost property is part of Knuth's proof: \begin{quote} Let $V$ be an internal node of maximum distance from the root. If $w_1$ and $w_2$ are not the weights already attached to the children of $V$, we can interchange them with the values that are already there; such an interchange does not increase the weighted path length. \end{quote} \noindent -Lemma~16.2 in Cormen et al.~\<^cite>\\p.~389\ in "cormen-et-al-2001"\ expresses a +Lemma~16.2 in Cormen et al.~\cite[p.~389]{cormen-et-al-2001} expresses a similar property, which turns out to be a corollary of our cost property: \begin{quote} Let $C$ be an alphabet in which each character $c \in C$ has frequency $f[c]$. Let $x$ and $y$ be two characters in $C$ having the lowest frequencies. Then there exists an optimal prefix code for $C$ in which the codewords for $x$ and $y$ have the same length and differ only in the last bit. \end{quote} \vskip-.75\myskipamount \ lemma cost_swapFourSyms_le: assumes "consistent t" "minima t a b" "c \ alphabet t" "d \ alphabet t" "depth t c = height t" "depth t d = height t" "c \ d" shows "cost (swapFourSyms t a b c d) \ cost t" proof - note lems = swapFourSyms_def minima_def cost_swapSyms_le depth_le_height show ?thesis proof (cases "a \ d \ b \ c") case True show ?thesis proof cases assume "a = c" show ?thesis proof cases assume "b = d" thus ?thesis using \a = c\ True assms by (simp add: lems) next assume "b \ d" thus ?thesis using \a = c\ True assms by (simp add: lems) qed next assume "a \ c" show ?thesis proof cases assume "b = d" thus ?thesis using \a \ c\ True assms by (simp add: lems) next assume "b \ d" have "cost (swapFourSyms t a b c d) \ cost (swapSyms t a c)" using \b \ d\ \a \ c\ True assms by (clarsimp simp: lems) also have "\ \ cost t" using \b \ d\ \a \ c\ True assms by (clarsimp simp: lems) finally show ?thesis . qed qed next case False thus ?thesis using assms by (clarsimp simp: lems) qed qed subsection \Leaf Split Optimality Lemma \label{leaf-split-optimality}\ text \ The tree @{term "splitLeaf t w\<^sub>a a w\<^sub>b b"} is optimum if $t$ is optimum, under a few assumptions, notably that $a$ and $b$ are minima of the new tree and that @{prop "freq t a = w\<^sub>a + w\<^sub>b"}. Graphically:\strut % \setbox\myboxi=\hbox{\includegraphics[scale=1.2]{tree-splitLeaf-a.pdf}}% \setbox\myboxii=\hbox{\includegraphics[scale=1.2]{tree-splitLeaf-ab.pdf}}% $${\it optimum\/} \smash{\lower\ht\myboxi\hbox{\raise.5\ht\myboxii\box\myboxi}} \,\mathop{\Longrightarrow}\;\;\; {\it optimum\/} \vcenter{\box\myboxii}$$ % This corresponds to the following fragment of Knuth's proof: \begin{quote} Now it is easy to prove that the weighted path length of such a tree is minimized if and only if the tree with $$\vcenter{\hbox{\includegraphics[scale=1.25]{tree-w1-w2-leaves.pdf}}} \qquad \hbox{replaced by} \qquad \vcenter{\hbox{\includegraphics[scale=1.25]{tree-w1-plus-w2.pdf}}}$$ has minimum path length for the weights $w_1 + w_2$, $w_3$, $\ldots\,$, $w_m$. \end{quote} \noindent We only need the ``if'' direction of Knuth's equivalence. Lemma~16.3 in -Cormen et al.~\<^cite>\\p.~391\ in "cormen-et-al-2001"\ expresses essentially the same +Cormen et al.~\cite[p.~391]{cormen-et-al-2001} expresses essentially the same property: \begin{quote} Let $C$ be a given alphabet with frequency $f[c]$ defined for each character $c \in C$. Let $x$ and $y$ be two characters in $C$ with minimum frequency. Let $C'$ be the alphabet $C$ with characters $x$, $y$ removed and (new) character $z$ added, so that $C' = C - \{x, y\} \cup {\{z\}}$; define $f$ for $C'$ as for $C$, except that $f[z] = f[x] + f[y]$. Let $T'$ be any tree representing an optimal prefix code for the alphabet $C'$. Then the tree $T$, obtained from $T'$ by replacing the leaf node for $z$ with an internal node having $x$ and $y$ as children, represents an optimal prefix code for the alphabet $C$. \end{quote} \noindent The proof is as follows: We assume that $t$ has a cost less than or equal to that of any other comparable tree~$v$ and show that @{term "splitLeaf t w\<^sub>a a w\<^sub>b b"} has a cost less than or equal to that of any other comparable tree $u$. By @{thm [source] exists_at_height} and @{thm [source] depth_height_imp_sibling_ne}, we know that some symbols $c$ and $d$ appear in sibling nodes at the very bottom of~$u$: $$\includegraphics[scale=1.25]{tree-splitLeaf-cd.pdf}$$ (The question mark is there to remind us that we know nothing specific about $u$'s structure.) From $u$ we construct a new tree @{term "swapFourSyms u a b c d"} in which the minima $a$ and $b$ are siblings: $$\includegraphics[scale=1.25]{tree-splitLeaf-abcd.pdf}$$ Merging $a$ and $b$ gives a tree comparable with $t$, which we can use to instantiate $v$ in the assumption: $$\includegraphics[scale=1.25]{tree-splitLeaf-abcd-aba.pdf}$$ With this instantiation, the proof is easy: $$\begin{tabularx}{\textwidth}{@% {\hskip\leftmargin}cX@% {}} & @{term "cost (splitLeaf t a w\<^sub>a b w\<^sub>b)"} \\ \eq & \justif{@{thm [source] cost_splitLeaf}} \\ & @{term "cost t + w\<^sub>a + w\<^sub>b"} \\ \kern-1em$\leq$\kern-1em & \justif{assumption} \\[-2ex] & $\cost (\ \overbrace{\strut\!@{term "mergeSibling (swapFourSyms u a b c d) a"}\!} ^{\smash{\hbox{$v$}}}\) + w\<^sub>a + w\<^sub>b\$ \\[\extrah] \eq & \justif{@{thm [source] cost_mergeSibling}} \\ & @{term "cost (swapFourSyms u a b c d)"} \\ \kern-1em$\leq$\kern-1em & \justif{@{thm [source] cost_swapFourSyms_le}} \\ & @{term "cost u"}. \\ \end{tabularx}$$ \noindent In contrast, the proof in Cormen et al.\ is by contradiction: Essentially, they assume that there exists a tree $u$ with a lower cost than @{term "splitLeaf t a w\<^sub>a b w\<^sub>b"} and show that there exists a tree~$v$ with a lower cost than~$t$, contradicting the hypothesis that $t$ is optimum. In place of @{thm [source] cost_swapFourSyms_le}, they invoke their lemma~16.2, which is questionable since $u$ is not necessarily optimum.% \footnote{Thomas Cormen commented that this step will be clarified in the next edition of the book.} Our proof relies on the following lemma, which asserts that $a$ and $b$ are minima of $u$. \ lemma twice_freq_le_imp_minima: "\\c \ alphabet t. w\<^sub>a \ freq t c \ w\<^sub>b \ freq t c; alphabet u = alphabet t \ {b}; a \ alphabet u; a \ b; freq u = (\c. if c = a then w\<^sub>a else if c = b then w\<^sub>b else freq t c)\ \ minima u a b" by (simp add: minima_def) text \ Now comes the key lemma. \ lemma optimum_splitLeaf: assumes "consistent t" "optimum t" "a \ alphabet t" "b \ alphabet t" "freq t a = w\<^sub>a + w\<^sub>b" "\c \ alphabet t. freq t c \ w\<^sub>a \ freq t c \ w\<^sub>b" shows "optimum (splitLeaf t w\<^sub>a a w\<^sub>b b)" proof (unfold optimum_def, clarify) fix u let ?t' = "splitLeaf t w\<^sub>a a w\<^sub>b b" assume c\<^sub>u: "consistent u" and a\<^sub>u: "alphabet ?t' = alphabet u" and f\<^sub>u: "freq ?t' = freq u" show "cost ?t' \ cost u" proof (cases "height ?t' = 0") case True thus ?thesis by simp next case False hence h\<^sub>u: "height u > 0" using a\<^sub>u assms by (auto intro: height_gt_0_alphabet_eq_imp_height_gt_0) have a\<^sub>a: "a \ alphabet u" using a\<^sub>u assms by fastforce have a\<^sub>b: "b \ alphabet u" using a\<^sub>u assms by fastforce have ab: "a \ b" using assms by blast from exists_at_height[OF c\<^sub>u] obtain c where a\<^sub>c: "c \ alphabet u" and d\<^sub>c: "depth u c = height u" .. let ?d = "sibling u c" have dc: "?d \ c" using d\<^sub>c c\<^sub>u h\<^sub>u a\<^sub>c by (metis depth_height_imp_sibling_ne) have a\<^sub>d: "?d \ alphabet u" using dc by (rule sibling_ne_imp_sibling_in_alphabet) have d\<^sub>d: "depth u ?d = height u" using d\<^sub>c c\<^sub>u by simp let ?u' = "swapFourSyms u a b c ?d" have c\<^sub>u\<^sub>': "consistent ?u'" using c\<^sub>u by simp have a\<^sub>u\<^sub>': "alphabet ?u' = alphabet u" using a\<^sub>a a\<^sub>b a\<^sub>c a\<^sub>d a\<^sub>u by simp have f\<^sub>u\<^sub>': "freq ?u' = freq u" using a\<^sub>a a\<^sub>b a\<^sub>c a\<^sub>d c\<^sub>u f\<^sub>u by simp have s\<^sub>a: "sibling ?u' a = b" using c\<^sub>u a\<^sub>a a\<^sub>b a\<^sub>c ab dc by (rule sibling_swapFourSyms_when_4th_is_sibling) let ?v = "mergeSibling ?u' a" have c\<^sub>v: "consistent ?v" using c\<^sub>u\<^sub>' by simp have a\<^sub>v: "alphabet ?v = alphabet t" using s\<^sub>a c\<^sub>u\<^sub>' a\<^sub>u\<^sub>' a\<^sub>a a\<^sub>u assms by auto have f\<^sub>v: "freq ?v = freq t" using s\<^sub>a c\<^sub>u\<^sub>' a\<^sub>u\<^sub>' f\<^sub>u\<^sub>' f\<^sub>u[THEN sym] ab a\<^sub>u[THEN sym] assms by (simp add: freq_mergeSibling ext) have "cost ?t' = cost t + w\<^sub>a + w\<^sub>b" using assms by simp also have "\ \ cost ?v + w\<^sub>a + w\<^sub>b" using c\<^sub>v a\<^sub>v f\<^sub>v \optimum t\ by (simp add: optimum_def) also have "\ = cost ?u'" proof - have "cost ?v + freq ?u' a + freq ?u' (sibling ?u' a) = cost ?u'" using c\<^sub>u\<^sub>' s\<^sub>a assms by (subst cost_mergeSibling) auto moreover have "w\<^sub>a = freq ?u' a" "w\<^sub>b = freq ?u' b" using f\<^sub>u\<^sub>' f\<^sub>u[THEN sym] assms by clarsimp+ ultimately show ?thesis using s\<^sub>a by simp qed also have "\ \ cost u" proof - have "minima u a b" using a\<^sub>u f\<^sub>u assms by (subst twice_freq_le_imp_minima) auto with c\<^sub>u show ?thesis using a\<^sub>c a\<^sub>d d\<^sub>c d\<^sub>d dc[THEN not_sym] by (rule cost_swapFourSyms_le) qed finally show ?thesis . qed qed subsection \Leaf Split Commutativity Lemma \label{leaf-split-commutativity}\ text \ A key property of Huffman's algorithm is that once it has combined two lowest-weight trees using @{const uniteTrees}, it does not visit these trees ever again. This suggests that splitting a leaf node before applying the algorithm should give the same result as applying the algorithm first and splitting the leaf node afterward. The diagram below illustrates the situation:\strut \def\myscale{1.05}% \setbox\myboxi=\hbox{(9)\strut}% \setbox\myboxii=\hbox{\includegraphics[scale=\myscale]{forest-a.pdf}}% $$(1)\,\lower\ht\myboxii\hbox{\raise\ht\myboxi\box\myboxii}$$ \smallskip \setbox\myboxii=\hbox{\includegraphics[scale=\myscale]{tree-splitLeaf-a.pdf}}% \setbox\myboxiii=\hbox{\includegraphics[scale=\myscale]% {forest-splitLeaf-ab.pdf}}% \mydimeni=\wd\myboxii \noindent (2a)\,\lower\ht\myboxii\hbox{\raise\ht\myboxi\box\myboxii}% \qquad\qquad\quad (2b)\,\lower\ht\myboxiii\hbox{\raise\ht\myboxi\box\myboxiii}\quad{} \setbox\myboxiii=\hbox{\includegraphics[scale=\myscale]% {tree-splitLeaf-ab.pdf}}% \setbox\myboxiv=\hbox{\includegraphics[scale=\myscale]% {tree-huffman-splitLeaf-ab.pdf}}% \mydimenii=\wd\myboxiii \vskip1.5\smallskipamount \noindent (3a)\,\lower\ht\myboxiii\hbox{\raise\ht\myboxi\box\myboxiii}% \qquad\qquad\quad (3b)\,\hfill\lower\ht\myboxiv\hbox{\raise\ht\myboxi\box\myboxiv}% \quad\hfill{} \noindent From the original forest (1), we can either run the algorithm (2a) and then split $a$ (3a) or split $a$ (2b) and then run the algorithm (3b). Our goal is to show that trees (3a) and (3b) are identical. Formally, we prove that $$@{term "splitLeaf (huffman ts) w\<^sub>a a w\<^sub>b b"} = @{term "huffman (splitLeaf\<^sub>F ts w\<^sub>a a w\<^sub>b b)"}$$ when @{term ts} is consistent, @{term "a \ alphabet\<^sub>F ts"}, @{term "b \ alphabet\<^sub>F ts"}, and $@{term "freq\<^sub>F ts a"} = @{term w\<^sub>a} \mathbin{\+\} @{term "w\<^sub>b"}$. But before we can prove this commutativity lemma, we need to introduce a few simple lemmas. \ lemma cachedWeight_splitLeaf[simp]: "cachedWeight (splitLeaf t w\<^sub>a a w\<^sub>b b) = cachedWeight t" by (case_tac t) simp+ lemma splitLeaf\<^sub>F_insortTree_when_in_alphabet_left[simp]: "\a \ alphabet t; consistent t; a \ alphabet\<^sub>F ts; freq t a = w\<^sub>a + w\<^sub>b\ \ splitLeaf\<^sub>F (insortTree t ts) w\<^sub>a a w\<^sub>b b = insortTree (splitLeaf t w\<^sub>a a w\<^sub>b b) ts" by (induct ts) simp+ lemma splitLeaf\<^sub>F_insortTree_when_in_alphabet\<^sub>F_tail[simp]: "\a \ alphabet\<^sub>F ts; consistent\<^sub>F ts; a \ alphabet t; freq\<^sub>F ts a = w\<^sub>a + w\<^sub>b\ \ splitLeaf\<^sub>F (insortTree t ts) w\<^sub>a a w\<^sub>b b = insortTree t (splitLeaf\<^sub>F ts w\<^sub>a a w\<^sub>b b)" proof (induct ts) case Nil thus ?case by simp next case (Cons u us) show ?case proof (cases "a \ alphabet u") case True hence "a \ alphabet\<^sub>F us" using Cons by auto thus ?thesis using Cons by auto next case False thus ?thesis using Cons by simp qed qed text \ We are now ready to prove the commutativity lemma. \ lemma splitLeaf_huffman_commute: "\consistent\<^sub>F ts; ts \ []; a \ alphabet\<^sub>F ts; freq\<^sub>F ts a = w\<^sub>a + w\<^sub>b\ \ splitLeaf (huffman ts) w\<^sub>a a w\<^sub>b b = huffman (splitLeaf\<^sub>F ts w\<^sub>a a w\<^sub>b b)" proof (induct ts rule: huffman.induct) case (1 t) thus ?case by simp next case (2 t\<^sub>1 t\<^sub>2 ts) note hyps = 2 show ?case proof (cases "a \ alphabet t\<^sub>1") case True hence "a \ alphabet t\<^sub>2" "a \ alphabet\<^sub>F ts" using hyps by auto thus ?thesis using hyps by (simp add: uniteTrees_def) next case False note a\<^sub>1 = False show ?thesis proof (cases "a \ alphabet t\<^sub>2") case True hence "a \ alphabet\<^sub>F ts" using hyps by auto thus ?thesis using a\<^sub>1 hyps by (simp add: uniteTrees_def) next case False thus ?thesis using a\<^sub>1 hyps by simp qed qed next case 3 thus ?case by simp qed text \ An important consequence of the commutativity lemma is that applying Huffman's algorithm on a forest of the form $$\vcenter{\hbox{\includegraphics[scale=1.25]{forest-uniteTrees.pdf}}}$$ gives the same result as applying the algorithm on the ``flat'' forest $$\vcenter{\hbox{\includegraphics[scale=1.25]{forest-uniteTrees-flat.pdf}}}$$ followed by splitting the leaf node $a$ into two nodes $a$, $b$ with frequencies $@{term w\<^sub>a}$, $@{term w\<^sub>b}$. The lemma effectively provides a way to flatten the forest at each step of the algorithm. Its invocation is implicit in the textbook proof. \ subsection \Optimality Theorem\ text \ We are one lemma away from our main result. \ lemma max_0_imp_0[simp]: "(max x y = (0::nat)) = (x = 0 \ y = 0)" by auto theorem optimum_huffman: "\consistent\<^sub>F ts; height\<^sub>F ts = 0; sortedByWeight ts; ts \ []\ \ optimum (huffman ts)" txt \ The input @{term ts} is assumed to be a nonempty consistent list of leaf nodes sorted by weight. The proof is by induction on the length of the forest @{term ts}. Let @{term ts} be $$\vcenter{\hbox{\includegraphics[scale=1.25]{forest-flat.pdf}}}$$ with $w_a \le w_b \le w_c \le w_d \le \cdots \le w_z$. If @{term ts} consists of a single leaf node, the node has cost 0 and is therefore optimum. If @{term ts} has length 2 or more, the first step of the algorithm leaves us with the term $${\it huffman\/}\enskip\; \vcenter{\hbox{\includegraphics[scale=1.25]% {forest-uniteTrees.pdf}}}$$ In the diagram, we put the newly created tree at position 2 in the forest; in general, it could be anywhere. By @{thm [source] splitLeaf_huffman_commute}, the above tree equals\strut $${\it splitLeaf\/}\;\left({\it huffman\/}\enskip\; \vcenter{\hbox{\includegraphics[scale=1.25]{forest-uniteTrees-flat.pdf}}} \;\right)\,\w\<^sub>a a w\<^sub>b b\.$$ To prove that this tree is optimum, it suffices by @{thm [source] optimum_splitLeaf} to show that\strut $${\it huffman\/}\enskip\; \vcenter{\hbox{\includegraphics[scale=1.25]{forest-uniteTrees-flat.pdf}}}$$ is optimum, which follows from the induction hypothesis.\strut \ proof (induct ts rule: length_induct) \ \\sc Complete induction step\ case (1 ts) note hyps = 1 show ?case proof (cases ts) case Nil thus ?thesis using \ts \ []\ by fast next case (Cons t\<^sub>a ts') note ts = Cons show ?thesis proof (cases ts') case Nil thus ?thesis using ts hyps by (simp add: optimum_def) next case (Cons t\<^sub>b ts'') note ts' = Cons show ?thesis proof (cases t\<^sub>a) case (Leaf w\<^sub>a a) note l\<^sub>a = Leaf show ?thesis proof (cases t\<^sub>b) case (Leaf w\<^sub>b b) note l\<^sub>b = Leaf show ?thesis proof - let ?us = "insortTree (uniteTrees t\<^sub>a t\<^sub>b) ts''" let ?us' = "insortTree (Leaf (w\<^sub>a + w\<^sub>b) a) ts''" let ?t\<^sub>s = "splitLeaf (huffman ?us') w\<^sub>a a w\<^sub>b b" have e\<^sub>1: "huffman ts = huffman ?us" using ts' ts by simp have e\<^sub>2: "huffman ?us = ?t\<^sub>s" using l\<^sub>a l\<^sub>b ts' ts hyps by (auto simp: splitLeaf_huffman_commute uniteTrees_def) have "optimum (huffman ?us')" using l\<^sub>a ts' ts hyps by (drule_tac x = ?us' in spec) (auto dest: sortedByWeight_Cons_imp_sortedByWeight simp: sortedByWeight_insortTree) hence "optimum ?t\<^sub>s" using l\<^sub>a l\<^sub>b ts' ts hyps apply simp apply (rule optimum_splitLeaf) by (auto dest!: height\<^sub>F_0_imp_Leaf_freq\<^sub>F_in_set sortedByWeight_Cons_imp_forall_weight_ge) thus "optimum (huffman ts)" using e\<^sub>1 e\<^sub>2 by simp qed next case Node thus ?thesis using ts' ts hyps by simp qed next case Node thus ?thesis using ts' ts hyps by simp qed qed qed qed text \ \isakeyword{end} \myskip \noindent So what have we achieved? Assuming that our definitions really mean what we intend them to mean, we established that our functional implementation of Huffman's algorithm, when invoked properly, constructs a binary tree that represents an optimal prefix code for the specified alphabet and frequencies. -Using Isabelle's code generator \<^cite>\"haftmann-nipkow-2007"\, we can convert the +Using Isabelle's code generator \cite{haftmann-nipkow-2007}, we can convert the Isabelle code into Standard ML, OCaml, or Haskell and use it in a real application. As a side note, the @{thm [source] optimum_huffman} theorem assumes that the forest @{term ts} passed to @{const huffman} consists exclusively of leaf nodes. It is tempting to relax this restriction, by requiring instead that the forest @{term ts} has the lowest cost among forests of the same size. We would define optimality of a forest as follows: $$\begin{aligned}[t] @{prop "optimum\<^sub>F ts"}\,\;\=\\;\, (\\us.\\ & \length ts = length us \ consistent\<^sub>F us \\ \\[-2.5pt] & \alphabet\<^sub>F ts = alphabet\<^sub>F us \ freq\<^sub>F ts = freq\<^sub>F us \\ \\[-2.5pt] & @{prop "cost\<^sub>F ts \ cost\<^sub>F us"})\end{aligned}$$ with $\cost\<^sub>F [] = 0\$ and $@{prop "cost\<^sub>F (t # ts) = cost t + cost\<^sub>F ts"}$. However, the modified proposition does not hold. A counterexample is the optimum forest $$\includegraphics{forest-optimal.pdf}$$ for which the algorithm constructs the tree $$\vcenter{\hbox{\includegraphics{tree-suboptimal.pdf}}} \qquad \hbox{of greater cost than} \qquad \vcenter{\hbox{\includegraphics{tree-optimal.pdf}}}$$ \ section \Related Work \label{related-work}\ text \ -Laurent Th\'ery's Coq formalization of Huffman's algorithm \<^cite>\"thery-2003" and "% -thery-2004"\ is an obvious yardstick for our work. It has a somewhat wider +Laurent Th\'ery's Coq formalization of Huffman's algorithm \cite{thery-2003, +thery-2004} is an obvious yardstick for our work. It has a somewhat wider scope, proving among others the isomorphism between prefix codes and full binary trees. With 291 theorems, it is also much larger. Th\'ery identified the following difficulties in formalizing the textbook proof: \begin{enumerate} \item The leaf interchange process that brings the two minimal symbols together is tedious to formalize. \item The sibling merging process requires introducing a new symbol for the merged node, which complicates the formalization. \item The algorithm constructs the tree in a bottom-up fashion. While top-down procedures can usually be proved by structural induction, bottom-up procedures often require more sophisticated induction principles and larger invariants. \item The informal proof relies on the notion of depth of a node. Defining this notion formally is problematic, because the depth can only be seen as a function if the tree is composed of distinct nodes. \end{enumerate} To circumvent these difficulties, Th\'ery introduced the ingenious concept of cover. A forest @{term ts} is a {\em cover\/} of a tree~$t$ if $t$ can be built from @{term ts} by adding inner nodes on top of the trees in @{term ts}. The term ``cover'' is easier to understand if the binary trees are drawn with the root at the bottom of the page, like natural trees. Huffman's algorithm is a refinement of the cover concept. The main proof consists in showing that the cost of @{term "huffman ts"} is less than or equal to that of any other tree for which @{term ts} is a cover. It relies on a few auxiliary definitions, notably an ``ordered cover'' concept that facilitates structural induction and a four-argument depth predicate (confusingly called @{term height}). Permutations also play a central role. Incidentally, our experience suggests that the potential problems identified by Th\'ery can be overcome more directly without too much work, leading to a simpler proof: \begin{enumerate} \item Formalizing the leaf interchange did not prove overly tedious. Among our 95~lemmas and theorems, 24 concern @{const swapLeaves}, @{const swapSyms}, and @{const swapFourSyms}. \item The generation of a new symbol for the resulting node when merging two sibling nodes in @{const mergeSibling} was trivially solved by reusing one of the two merged symbols. \item The bottom-up nature of the tree construction process was addressed by using the length of the forest as the induction measure and by merging the two minimal symbols, as in Knuth's proof. \item By restricting our attention to consistent trees, we were able to define the @{const depth} function simply and meaningfully. \end{enumerate} \ section \Conclusion \label{conclusion}\ text \ The goal of most formal proofs is to increase our confidence in a result. In the case of Huffman's algorithm, however, the chances that a bug would have gone unnoticed for the 56 years since its publication, under the scrutiny of leading computer scientists, seem extremely low; and the existence of a Coq proof should be sufficient to remove any remaining doubts. The main contribution of this document has been to demonstrate that the textbook proof of Huffman's algorithm can be elegantly formalized using a state-of-the-art theorem prover such as Isabelle/HOL. In the process, we uncovered a few minor snags in the proof given in Cormen et -al.~\<^cite>\"cormen-et-al-2001"\. +al.~\cite{cormen-et-al-2001}. We also found that custom induction rules, in combination with suitable simplification rules, greatly help the automatic proof tactics, sometimes reducing 30-line proof scripts to one-liners. We successfully applied this approach for handling both the ubiquitous ``datatype + well\-formed\-ness predicate'' combination (@{typ "'a tree"} + @{const consistent}) and functions defined by sequential pattern matching (@{const sibling} and @{const mergeSibling}). Our experience suggests that such rules, which are uncommon in formalizations, are highly valuable and versatile. Moreover, Isabelle's \textit{induction\_schema} and \textit{lexicographic\_order} tactics make these easy to prove. Formalizing the proof of Huffman's algorithm also led to a deeper understanding of this classic algorithm. Many of the lemmas, notably the leaf split commutativity lemma of Section~\ref{leaf-split-commutativity}, have not been found in the literature and express fundamental properties of the algorithm. Other discoveries did not find their way into the final proof. In particular, each step of the algorithm appears to preserve the invariant that the nodes in a forest are ordered by weight from left to right, bottom to top, as in the example below:\strut $$\vcenter{\hbox{\includegraphics[scale=1.25]{forest-zigzag.pdf}}}$$ It is not hard to prove formally that a tree exhibiting this property is optimum. On the other hand, proving that the algorithm preserves this invariant seems difficult---more difficult than formalizing the textbook proof---and remains a suggestion for future work. A few other directions for future work suggest themselves. First, we could formalize some of our hypotheses, notably our restriction to full and consistent binary trees. The current formalization says nothing about the algorithm's application for data compression, so the next step could be to extend the proof's scope to cover @{term encode}/@{term decode} functions and show that full binary trees are isomorphic to prefix codes, as done in the Coq development. Independently, we could generalize the development to $n$-ary trees. \ (*<*) end (*>*)