author | wenzelm |
Sun, 21 Oct 2001 19:35:40 +0200 | |
changeset 11858 | ca128c9100b6 |
child 11901 | e1aa90e4ef4e |
permissions | -rw-r--r-- |
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1 |
(*<*)theory Typedefs = Main:(*>*) |
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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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axioms |
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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. Thus |
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a type definition is merely a notational device: you introduce a new name for |
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a subset of an existing type. This does not add any logical power to HOL, |
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because you could base all your work directly on the subset of the existing |
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type. However, the resulting theories could easily become indigestible |
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because instead of implicit types you would have explicit sets in your |
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formulae. |
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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 = "{n::nat. n \<le> 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: take 0 as a witness. |
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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"{0,1,2}"}. 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"{0,1,2}"}. 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 three} &::& @{typ"nat set"} \\ |
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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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where constant @{term three} is explicitly defined as the representing set: |
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\begin{center} |
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@{thm three_def}\hfill(@{thm[source]three_def}) |
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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 are types and circles are sets: |
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\begin{center} |
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\unitlength1mm |
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\thicklines |
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\begin{picture}(100,40) |
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\put(3,13){\framebox(15,15){@{typ three}}} |
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\put(55,5){\framebox(30,30){@{term three}}} |
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\put(70,32){\makebox(0,0){@{typ nat}}} |
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\put(70,20){\circle{40}} |
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\put(10,15){\vector(1,0){60}} |
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\put(25,14){\makebox(0,0)[tl]{@{term Rep_three}}} |
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\put(70,25){\vector(-1,0){60}} |
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\put(25,26){\makebox(0,0)[bl]{@{term Abs_three}}} |
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119 |
\end{picture} |
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120 |
\end{center} |
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121 |
Finally, \isacommand{typedef} asserts that @{term Rep_three} is |
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122 |
surjective on the subset @{term three} and @{term Abs_three} and @{term |
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123 |
Rep_three} are inverses of each other: |
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124 |
\begin{center} |
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125 |
\begin{tabular}{@ {}r@ {\qquad\qquad}l@ {}} |
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126 |
@{thm Rep_three[no_vars]} & (@{thm[source]Rep_three}) \\ |
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127 |
@{thm Rep_three_inverse[no_vars]} & (@{thm[source]Rep_three_inverse}) \\ |
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128 |
@{thm Abs_three_inverse[no_vars]} & (@{thm[source]Abs_three_inverse}) |
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129 |
\end{tabular} |
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130 |
\end{center} |
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131 |
% |
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132 |
From this example it should be clear what \isacommand{typedef} does |
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133 |
in general given a name (here @{text three}) and a set |
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134 |
(here @{term"{n. n\<le>2}"}). |
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135 |
|
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136 |
Our next step is to define the basic functions expected on the new type. |
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137 |
Although this depends on the type at hand, the following strategy works well: |
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138 |
\begin{itemize} |
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\item define a small kernel of basic functions that can express all other |
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140 |
functions you anticipate. |
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141 |
\item define the kernel in terms of corresponding functions on the |
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142 |
representing type using @{term Abs} and @{term Rep} to convert between the |
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143 |
two levels. |
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144 |
\end{itemize} |
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145 |
In our example it suffices to give the three elements of type @{typ three} |
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146 |
names: |
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147 |
*} |
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148 |
|
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149 |
constdefs |
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150 |
A:: three |
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151 |
"A \<equiv> Abs_three 0" |
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152 |
B:: three |
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153 |
"B \<equiv> Abs_three 1" |
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154 |
C :: three |
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155 |
"C \<equiv> Abs_three 2" |
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156 |
|
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157 |
text{* |
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158 |
So far, everything was easy. But it is clear that reasoning about @{typ |
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159 |
three} will be hell if we have to go back to @{typ nat} every time. Thus our |
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160 |
aim must be to raise our level of abstraction by deriving enough theorems |
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161 |
about type @{typ three} to characterize it completely. And those theorems |
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162 |
should be phrased in terms of @{term A}, @{term B} and @{term C}, not @{term |
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163 |
Abs_three} and @{term Rep_three}. Because of the simplicity of the example, |
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164 |
we merely need to prove that @{term A}, @{term B} and @{term C} are distinct |
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165 |
and that they exhaust the type. |
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166 |
|
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167 |
In processing our \isacommand{typedef} declaration, |
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168 |
Isabelle helpfully proves several lemmas. |
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169 |
One, @{thm[source]Abs_three_inject}, |
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170 |
expresses that @{term Abs_three} is injective on the representing subset: |
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171 |
\begin{center} |
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172 |
@{thm Abs_three_inject[no_vars]} |
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173 |
\end{center} |
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174 |
Another, @{thm[source]Rep_three_inject}, expresses that the representation |
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175 |
function is also injective: |
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176 |
\begin{center} |
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177 |
@{thm Rep_three_inject[no_vars]} |
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178 |
\end{center} |
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179 |
Distinctness of @{term A}, @{term B} and @{term C} follows immediately |
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180 |
if we expand their definitions and rewrite with the injectivity |
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181 |
of @{term Abs_three}: |
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182 |
*} |
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183 |
|
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184 |
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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185 |
by(simp add: Abs_three_inject A_def B_def C_def three_def) |
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186 |
|
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187 |
text{*\noindent |
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188 |
Of course we rely on the simplifier to solve goals like @{prop"0 \<noteq> 1"}. |
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189 |
|
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190 |
The fact that @{term A}, @{term B} and @{term C} exhaust type @{typ three} is |
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191 |
best phrased as a case distinction theorem: if you want to prove @{prop"P x"} |
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192 |
(where @{term x} is of type @{typ three}) it suffices to prove @{prop"P A"}, |
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193 |
@{prop"P B"} and @{prop"P C"}. First we prove the analogous proposition for the |
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194 |
representation: *} |
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195 |
|
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196 |
lemma cases_lemma: "\<lbrakk> Q 0; Q 1; Q 2; n \<in> three \<rbrakk> \<Longrightarrow> Q n" |
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197 |
|
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198 |
txt{*\noindent |
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199 |
Expanding @{thm[source]three_def} yields the premise @{prop"n\<le>2"}. Repeated |
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200 |
elimination with @{thm[source]le_SucE} |
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201 |
@{thm[display]le_SucE} |
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202 |
reduces @{prop"n\<le>2"} to the three cases @{prop"n\<le>0"}, @{prop"n=1"} and |
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203 |
@{prop"n=2"} which are trivial for simplification: |
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204 |
*} |
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205 |
|
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206 |
apply(simp add: three_def numerals) |
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207 |
apply((erule le_SucE)+) |
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208 |
apply simp_all |
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209 |
done |
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210 |
|
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211 |
text{* |
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212 |
Now the case distinction lemma on type @{typ three} is easy to derive if you |
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213 |
know how: |
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214 |
*} |
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215 |
|
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216 |
lemma three_cases: "\<lbrakk> P A; P B; P C \<rbrakk> \<Longrightarrow> P x" |
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217 |
|
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218 |
txt{*\noindent |
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219 |
We start by replacing the @{term x} by @{term"Abs_three(Rep_three x)"}: |
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220 |
*} |
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221 |
|
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222 |
apply(rule subst[OF Rep_three_inverse]) |
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223 |
|
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224 |
txt{*\noindent |
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225 |
This substitution step worked nicely because there was just a single |
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226 |
occurrence of a term of type @{typ three}, namely @{term x}. |
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227 |
When we now apply @{thm[source]cases_lemma}, @{term Q} becomes @{term"\<lambda>n. P(Abs_three |
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228 |
n)"} because @{term"Rep_three x"} is the only term of type @{typ nat}: |
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229 |
*} |
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230 |
|
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231 |
apply(rule cases_lemma) |
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232 |
|
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233 |
txt{* |
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234 |
@{subgoals[display,indent=0]} |
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235 |
The resulting subgoals are easily solved by simplification: |
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236 |
*} |
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237 |
|
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apply(simp_all add:A_def B_def C_def Rep_three) |
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done |
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|
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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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|
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datatype three' = A | B | C |
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|
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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 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(*>*) |