author | wenzelm |
Fri, 12 Oct 2012 18:58:20 +0200 | |
changeset 49834 | b27bbb021df1 |
parent 49812 | e3945ddcb9aa |
child 58620 | 7435b6a3f72e |
permissions | -rw-r--r-- |
17914 | 1 |
(*<*)theory Typedefs imports Main begin(*>*) |
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section{*Introducing New Types*} |
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text{*\label{sec:adv-typedef} |
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For most applications, a combination of predefined types like @{typ bool} and |
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@{text"\<Rightarrow>"} with recursive datatypes and records is quite sufficient. Very |
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occasionally you may feel the need for a more advanced type. If you |
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are certain that your type is not definable by any of the |
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standard means, then read on. |
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\begin{warn} |
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Types in HOL must be non-empty; otherwise the quantifier rules would be |
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unsound, because $\exists x.\ x=x$ is a theorem. |
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\end{warn} |
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*} |
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subsection{*Declaring New Types*} |
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text{*\label{sec:typedecl} |
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\index{types!declaring|(}% |
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\index{typedecl@\isacommand {typedecl} (command)}% |
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The most trivial way of introducing a new type is by a \textbf{type |
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declaration}: *} |
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typedecl my_new_type |
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text{*\noindent |
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This does not define @{typ my_new_type} at all but merely introduces its |
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name. Thus we know nothing about this type, except that it is |
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non-empty. Such declarations without definitions are |
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useful if that type can be viewed as a parameter of the theory. |
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A typical example is given in \S\ref{sec:VMC}, where we define a transition |
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relation over an arbitrary type of states. |
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In principle we can always get rid of such type declarations by making those |
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types parameters of every other type, thus keeping the theory generic. In |
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practice, however, the resulting clutter can make types hard to read. |
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If you are looking for a quick and dirty way of introducing a new type |
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together with its properties: declare the type and state its properties as |
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axioms. Example: |
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*} |
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axiomatization where |
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just_one: "\<exists>x::my_new_type. \<forall>y. x = y" |
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text{*\noindent |
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However, we strongly discourage this approach, except at explorative stages |
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of your development. It is extremely easy to write down contradictory sets of |
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axioms, in which case you will be able to prove everything but it will mean |
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nothing. In the example above, the axiomatic approach is |
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unnecessary: a one-element type called @{typ unit} is already defined in HOL. |
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\index{types!declaring|)} |
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*} |
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subsection{*Defining New Types*} |
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text{*\label{sec:typedef} |
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\index{types!defining|(}% |
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\index{typedecl@\isacommand {typedef} (command)|(}% |
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Now we come to the most general means of safely introducing a new type, the |
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\textbf{type definition}. All other means, for example |
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\isacommand{datatype}, are based on it. The principle is extremely simple: |
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any non-empty subset of an existing type can be turned into a new type. |
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More precisely, the new type is specified to be isomorphic to some |
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non-empty subset of an existing type. |
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Let us work a simple example, the definition of a three-element type. |
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It is easily represented by the first three natural numbers: |
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*} |
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typedef three = "{0::nat, 1, 2}" |
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txt{*\noindent |
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In order to enforce that the representing set on the right-hand side is |
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non-empty, this definition actually starts a proof to that effect: |
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@{subgoals[display,indent=0]} |
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Fortunately, this is easy enough to show, even \isa{auto} could do it. |
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In general, one has to provide a witness, in our case 0: |
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*} |
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apply(rule_tac x = 0 in exI) |
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by simp |
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text{* |
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This type definition introduces the new type @{typ three} and asserts |
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that it is a copy of the set @{term"{0::nat,1,2}"}. This assertion |
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is expressed via a bijection between the \emph{type} @{typ three} and the |
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\emph{set} @{term"{0::nat,1,2}"}. To this end, the command declares the following |
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constants behind the scenes: |
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\begin{center} |
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\begin{tabular}{rcl} |
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@{term Rep_three} &::& @{typ"three \<Rightarrow> nat"}\\ |
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@{term Abs_three} &::& @{typ"nat \<Rightarrow> three"} |
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\end{tabular} |
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\end{center} |
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The situation is best summarized with the help of the following diagram, |
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where squares denote types and the irregular region denotes a set: |
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\begin{center} |
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\includegraphics[scale=.8]{typedef} |
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\end{center} |
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Finally, \isacommand{typedef} asserts that @{term Rep_three} is |
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surjective on the subset and @{term Abs_three} and @{term |
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Rep_three} are inverses of each other: |
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\begin{center} |
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\begin{tabular}{@ {}r@ {\qquad\qquad}l@ {}} |
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@{thm Rep_three[no_vars]} & (@{thm[source]Rep_three}) \\ |
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@{thm Rep_three_inverse[no_vars]} & (@{thm[source]Rep_three_inverse}) \\ |
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@{thm Abs_three_inverse[no_vars]} & (@{thm[source]Abs_three_inverse}) |
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\end{tabular} |
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\end{center} |
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% |
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From this example it should be clear what \isacommand{typedef} does |
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in general given a name (here @{text three}) and a set |
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(here @{term"{0::nat,1,2}"}). |
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Our next step is to define the basic functions expected on the new type. |
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Although this depends on the type at hand, the following strategy works well: |
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\begin{itemize} |
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\item define a small kernel of basic functions that can express all other |
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functions you anticipate. |
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\item define the kernel in terms of corresponding functions on the |
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representing type using @{term Abs} and @{term Rep} to convert between the |
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two levels. |
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\end{itemize} |
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In our example it suffices to give the three elements of type @{typ three} |
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names: |
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*} |
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definition A :: three where "A \<equiv> Abs_three 0" |
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definition B :: three where "B \<equiv> Abs_three 1" |
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definition C :: three where "C \<equiv> Abs_three 2" |
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text{* |
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So far, everything was easy. But it is clear that reasoning about @{typ |
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three} will be hell if we have to go back to @{typ nat} every time. Thus our |
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aim must be to raise our level of abstraction by deriving enough theorems |
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about type @{typ three} to characterize it completely. And those theorems |
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should be phrased in terms of @{term A}, @{term B} and @{term C}, not @{term |
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Abs_three} and @{term Rep_three}. Because of the simplicity of the example, |
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we merely need to prove that @{term A}, @{term B} and @{term C} are distinct |
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and that they exhaust the type. |
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|
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In processing our \isacommand{typedef} declaration, |
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Isabelle proves several helpful lemmas. The first two |
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express injectivity of @{term Rep_three} and @{term Abs_three}: |
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\begin{center} |
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\begin{tabular}{@ {}r@ {\qquad}l@ {}} |
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@{thm Rep_three_inject[no_vars]} & (@{thm[source]Rep_three_inject}) \\ |
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\begin{tabular}{@ {}l@ {}} |
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@{text"\<lbrakk>x \<in> {0, 1, 2}; y \<in> {0, 1, 2} \<rbrakk>"} \\ |
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@{text"\<Longrightarrow> (Abs_three x = Abs_three y) = (x = y)"} |
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\end{tabular} & (@{thm[source]Abs_three_inject}) \\ |
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\end{tabular} |
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\end{center} |
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The following ones allow to replace some @{text"x::three"} by |
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@{text"Abs_three(y::nat)"}, and conversely @{term y} by @{term"Rep_three x"}: |
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\begin{center} |
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\begin{tabular}{@ {}r@ {\qquad}l@ {}} |
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@{thm Rep_three_cases[no_vars]} & (@{thm[source]Rep_three_cases}) \\ |
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@{thm Abs_three_cases[no_vars]} & (@{thm[source]Abs_three_cases}) \\ |
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@{thm Rep_three_induct[no_vars]} & (@{thm[source]Rep_three_induct}) \\ |
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@{thm Abs_three_induct[no_vars]} & (@{thm[source]Abs_three_induct}) \\ |
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\end{tabular} |
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\end{center} |
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These theorems are proved for any type definition, with @{text three} |
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replaced by the name of the type in question. |
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Distinctness of @{term A}, @{term B} and @{term C} follows immediately |
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if we expand their definitions and rewrite with the injectivity |
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of @{term Abs_three}: |
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*} |
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lemma "A \<noteq> B \<and> B \<noteq> A \<and> A \<noteq> C \<and> C \<noteq> A \<and> B \<noteq> C \<and> C \<noteq> B" |
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by(simp add: Abs_three_inject A_def B_def C_def) |
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text{*\noindent |
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Of course we rely on the simplifier to solve goals like @{prop"(0::nat) \<noteq> 1"}. |
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The fact that @{term A}, @{term B} and @{term C} exhaust type @{typ three} is |
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best phrased as a case distinction theorem: if you want to prove @{prop"P x"} |
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(where @{term x} is of type @{typ three}) it suffices to prove @{prop"P A"}, |
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@{prop"P B"} and @{prop"P C"}: *} |
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lemma three_cases: "\<lbrakk> P A; P B; P C \<rbrakk> \<Longrightarrow> P x" |
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txt{*\noindent Again this follows easily using the induction principle stemming from the type definition:*} |
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apply(induct_tac x) |
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txt{* |
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@{subgoals[display,indent=0]} |
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Simplification leads to the disjunction @{prop"y |
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= 0 \<or> y = 1 \<or> y = (2::nat)"} which \isa{auto} separates into three |
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subgoals, each of which is easily solved by simplification: *} |
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apply(auto simp add: A_def B_def C_def) |
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done |
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text{*\noindent |
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This concludes the derivation of the characteristic theorems for |
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type @{typ three}. |
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|
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The attentive reader has realized long ago that the |
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above lengthy definition can be collapsed into one line: |
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*} |
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datatype better_three = A | B | C |
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text{*\noindent |
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In fact, the \isacommand{datatype} command performs internally more or less |
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the same derivations as we did, which gives you some idea what life would be |
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like without \isacommand{datatype}. |
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|
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Although @{typ three} could be defined in one line, we have chosen this |
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example to demonstrate \isacommand{typedef} because its simplicity makes the |
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key concepts particularly easy to grasp. If you would like to see a |
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non-trivial example that cannot be defined more directly, we recommend the |
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definition of \emph{finite multisets} in the Library~\cite{HOL-Library}. |
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|
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Let us conclude by summarizing the above procedure for defining a new type. |
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Given some abstract axiomatic description $P$ of a type $ty$ in terms of a |
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set of functions $F$, this involves three steps: |
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\begin{enumerate} |
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\item Find an appropriate type $\tau$ and subset $A$ which has the desired |
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properties $P$, and make a type definition based on this representation. |
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\item Define the required functions $F$ on $ty$ by lifting |
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analogous functions on the representation via $Abs_ty$ and $Rep_ty$. |
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\item Prove that $P$ holds for $ty$ by lifting $P$ from the representation. |
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\end{enumerate} |
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You can now forget about the representation and work solely in terms of the |
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abstract functions $F$ and properties $P$.% |
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\index{typedecl@\isacommand {typedef} (command)|)}% |
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\index{types!defining|)} |
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*} |
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|
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(*<*)end(*>*) |