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(*<*)
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theory Isar
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imports LaTeXsugar
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begin
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ML{* quick_and_dirty := true *}
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(*>*)
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text{*
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Apply-scripts are unreadable and hard to maintain. The language of choice
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for larger proofs is \concept{Isar}. The two key features of Isar are:
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\begin{itemize}
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\item It is structured, not linear.
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\item It is readable without running it because
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you need to state what you are proving at any given point.
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\end{itemize}
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Whereas apply-scripts are like assembly language programs, Isar proofs
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are like structured programs with comments. A typical Isar proof looks like this:
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*}text{*
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\begin{tabular}{@ {}l}
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\isacom{proof}\\
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\quad\isacom{assume} @{text"\""}$\mathit{formula}_0$@{text"\""}\\
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\quad\isacom{have} @{text"\""}$\mathit{formula}_1$@{text"\""} \quad\isacom{by} @{text simp}\\
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\quad\vdots\\
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\quad\isacom{have} @{text"\""}$\mathit{formula}_n$@{text"\""} \quad\isacom{by} @{text blast}\\
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\quad\isacom{show} @{text"\""}$\mathit{formula}_{n+1}$@{text"\""} \quad\isacom{by} @{text \<dots>}\\
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\isacom{qed}
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\end{tabular}
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*}text{*
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It proves $\mathit{formula}_0 \Longrightarrow \mathit{formula}_{n+1}$
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(provided each proof step succeeds).
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The intermediate \isacom{have} statements are merely stepping stones
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on the way towards the \isacom{show} statement that proves the actual
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goal. In more detail, this is the Isar core syntax:
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\medskip
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\begin{tabular}{@ {}lcl@ {}}
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\textit{proof} &=& \isacom{by} \textit{method}\\
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&$\mid$& \isacom{proof} [\textit{method}] \ \textit{step}$^*$ \ \isacom{qed}
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\end{tabular}
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\medskip
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\begin{tabular}{@ {}lcl@ {}}
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\textit{step} &=& \isacom{fix} \textit{variables} \\
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&$\mid$& \isacom{assume} \textit{proposition} \\
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&$\mid$& [\isacom{from} \textit{fact}$^+$] (\isacom{have} $\mid$ \isacom{show}) \ \textit{proposition} \ \textit{proof}
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\end{tabular}
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\medskip
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\begin{tabular}{@ {}lcl@ {}}
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\textit{proposition} &=& [\textit{name}:] @{text"\""}\textit{formula}@{text"\""}
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\end{tabular}
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\medskip
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\begin{tabular}{@ {}lcl@ {}}
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\textit{fact} &=& \textit{name} \ $\mid$ \ \dots
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\end{tabular}
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\medskip
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\noindent A proof can either be an atomic \isacom{by} with a single proof
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method which must finish off the statement being proved, for example @{text
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auto}. Or it can be a \isacom{proof}--\isacom{qed} block of multiple
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steps. Such a block can optionally begin with a proof method that indicates
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how to start off the proof, e.g.\ \mbox{@{text"(induction xs)"}}.
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A step either assumes a proposition or states a proposition
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together with its proof. The optional \isacom{from} clause
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indicates which facts are to be used in the proof.
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Intermediate propositions are stated with \isacom{have}, the overall goal
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with \isacom{show}. A step can also introduce new local variables with
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\isacom{fix}. Logically, \isacom{fix} introduces @{text"\<And>"}-quantified
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variables, \isacom{assume} introduces the assumption of an implication
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(@{text"\<Longrightarrow>"}) and \isacom{have}/\isacom{show} the conclusion.
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Propositions are optionally named formulas. These names can be referred to in
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later \isacom{from} clauses. In the simplest case, a fact is such a name.
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But facts can also be composed with @{text OF} and @{text of} as shown in
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\S\ref{sec:forward-proof}---hence the \dots\ in the above grammar. Note
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that assumptions, intermediate \isacom{have} statements and global lemmas all
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have the same status and are thus collectively referred to as
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\concept{facts}.
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Fact names can stand for whole lists of facts. For example, if @{text f} is
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defined by command \isacom{fun}, @{text"f.simps"} refers to the whole list of
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recursion equations defining @{text f}. Individual facts can be selected by
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writing @{text"f.simps(2)"}, whole sublists by @{text"f.simps(2-4)"}.
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\section{Isar by example}
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We show a number of proofs of Cantor's theorem that a function from a set to
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its powerset cannot be surjective, illustrating various features of Isar. The
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constant @{const surj} is predefined.
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*}
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lemma "\<not> surj(f :: 'a \<Rightarrow> 'a set)"
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proof
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assume 0: "surj f"
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from 0 have 1: "\<forall>A. \<exists>a. A = f a" by(simp add: surj_def)
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from 1 have 2: "\<exists>a. {x. x \<notin> f x} = f a" by blast
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from 2 show "False" by blast
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qed
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text{*
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The \isacom{proof} command lacks an explicit method how to perform
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the proof. In such cases Isabelle tries to use some standard introduction
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rule, in the above case for @{text"\<not>"}:
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\[
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\inferrule{
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\mbox{@{thm (prem 1) notI}}}
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{\mbox{@{thm (concl) notI}}}
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\]
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In order to prove @{prop"~ P"}, assume @{text P} and show @{text False}.
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Thus we may assume @{prop"surj f"}. The proof shows that names of propositions
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may be (single!) digits---meaningful names are hard to invent and are often
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not necessary. Both \isacom{have} steps are obvious. The second one introduces
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the diagonal set @{term"{x. x \<notin> f x}"}, the key idea in the proof.
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If you wonder why @{text 2} directly implies @{text False}: from @{text 2}
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it follows that @{prop"a \<notin> f a \<longleftrightarrow> a \<in> f a"}.
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\subsection{@{text this}, @{text then}, @{text hence} and @{text thus}}
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Labels should be avoided. They interrupt the flow of the reader who has to
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scan the context for the point where the label was introduced. Ideally, the
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proof is a linear flow, where the output of one step becomes the input of the
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next step, piping the previously proved fact into the next proof, just like
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in a UNIX pipe. In such cases the predefined name @{text this} can be used
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to refer to the proposition proved in the previous step. This allows us to
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eliminate all labels from our proof (we suppress the \isacom{lemma} statement):
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*}
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(*<*)
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lemma "\<not> surj(f :: 'a \<Rightarrow> 'a set)"
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(*>*)
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proof
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assume "surj f"
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from this have "\<exists>a. {x. x \<notin> f x} = f a" by(auto simp: surj_def)
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from this show "False" by blast
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qed
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text{* We have also taken the opportunity to compress the two \isacom{have}
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steps into one.
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To compact the text further, Isar has a few convenient abbreviations:
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\medskip
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\begin{tabular}{rcl}
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\isacom{then} &=& \isacom{from} @{text this}\\
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\isacom{thus} &=& \isacom{then} \isacom{show}\\
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\isacom{hence} &=& \isacom{then} \isacom{have}
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\end{tabular}
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\medskip
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\noindent
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With the help of these abbreviations the proof becomes
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*}
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(*<*)
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lemma "\<not> surj(f :: 'a \<Rightarrow> 'a set)"
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(*>*)
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proof
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assume "surj f"
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hence "\<exists>a. {x. x \<notin> f x} = f a" by(auto simp: surj_def)
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thus "False" by blast
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qed
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text{*
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There are two further linguistic variations:
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\medskip
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\begin{tabular}{rcl}
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(\isacom{have}$\mid$\isacom{show}) \ \textit{prop} \ \isacom{using} \ \textit{facts}
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&=&
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\isacom{from} \ \textit{facts} \ (\isacom{have}$\mid$\isacom{show}) \ \textit{prop}\\
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\isacom{with} \ \textit{facts} &=& \isacom{from} \ \textit{facts} \isa{this}
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\end{tabular}
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\medskip
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\noindent The \isacom{using} idiom de-emphasises the used facts by moving them
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behind the proposition.
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\subsection{Structured lemma statements: \isacom{fixes}, \isacom{assumes}, \isacom{shows}}
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Lemmas can also be stated in a more structured fashion. To demonstrate this
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feature with Cantor's theorem, we rephrase @{prop"\<not> surj f"}
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a little:
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*}
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lemma
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fixes f :: "'a \<Rightarrow> 'a set"
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assumes s: "surj f"
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shows "False"
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txt{* The optional \isacom{fixes} part allows you to state the types of
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variables up front rather than by decorating one of their occurrences in the
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formula with a type constraint. The key advantage of the structured format is
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the \isacom{assumes} part that allows you to name each assumption; multiple
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assumptions can be separated by \isacom{and}. The
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\isacom{shows} part gives the goal. The actual theorem that will come out of
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the proof is @{prop"surj f \<Longrightarrow> False"}, but during the proof the assumption
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@{prop"surj f"} is available under the name @{text s} like any other fact.
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*}
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proof -
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have "\<exists> a. {x. x \<notin> f x} = f a" using s
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by(auto simp: surj_def)
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thus "False" by blast
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qed
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text{* In the \isacom{have} step the assumption @{prop"surj f"} is now
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referenced by its name @{text s}. The duplication of @{prop"surj f"} in the
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above proofs (once in the statement of the lemma, once in its proof) has been
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eliminated.
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\begin{warn}
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Note the dash after the \isacom{proof}
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command. It is the null method that does nothing to the goal. Leaving it out
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would ask Isabelle to try some suitable introduction rule on the goal @{const
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False}---but there is no suitable introduction rule and \isacom{proof}
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would fail.
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\end{warn}
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Stating a lemma with \isacom{assumes}-\isacom{shows} implicitly introduces the
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name @{text assms} that stands for the list of all assumptions. You can refer
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to individual assumptions by @{text"assms(1)"}, @{text"assms(2)"} etc,
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thus obviating the need to name them individually.
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\section{Proof patterns}
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We show a number of important basic proof patterns. Many of them arise from
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the rules of natural deduction that are applied by \isacom{proof} by
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default. The patterns are phrased in terms of \isacom{show} but work for
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\isacom{have} and \isacom{lemma}, too.
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We start with two forms of \concept{case distinction}:
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starting from a formula @{text P} we have the two cases @{text P} and
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@{prop"~P"}, and starting from a fact @{prop"P \<or> Q"}
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we have the two cases @{text P} and @{text Q}:
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*}text_raw{*
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\begin{tabular}{@ {}ll@ {}}
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\begin{minipage}[t]{.4\textwidth}
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\isa{%
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*}
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(*<*)lemma "R" proof-(*>*)
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show "R"
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proof cases
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assume "P"
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txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
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show "R" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
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next
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assume "\<not> P"
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txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
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show "R" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
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qed(*<*)oops(*>*)
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text_raw {* }
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\end{minipage}
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&
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\begin{minipage}[t]{.4\textwidth}
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\isa{%
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*}
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(*<*)lemma "R" proof-(*>*)
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have "P \<or> Q" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
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then show "R"
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proof
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assume "P"
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txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
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show "R" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
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next
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assume "Q"
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txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
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show "R" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
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qed(*<*)oops(*>*)
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text_raw {* }
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\end{minipage}
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\end{tabular}
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\medskip
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\begin{isamarkuptext}%
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How to prove a logical equivalence:
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\end{isamarkuptext}%
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\isa{%
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*}
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(*<*)lemma "P\<longleftrightarrow>Q" proof-(*>*)
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show "P \<longleftrightarrow> Q"
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proof
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assume "P"
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txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
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show "Q" (*<*)sorry(*>*) txt_raw{*\ $\dots$\\*}
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next
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assume "Q"
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txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
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show "P" (*<*)sorry(*>*) txt_raw{*\ $\dots$\\*}
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qed(*<*)qed(*>*)
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text_raw {* }
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\medskip
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\begin{isamarkuptext}%
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Proofs by contradiction:
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\end{isamarkuptext}%
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\begin{tabular}{@ {}ll@ {}}
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\begin{minipage}[t]{.4\textwidth}
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\isa{%
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*}
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(*<*)lemma "\<not> P" proof-(*>*)
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show "\<not> P"
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proof
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assume "P"
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txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
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show "False" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
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qed(*<*)oops(*>*)
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text_raw {* }
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\end{minipage}
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&
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\begin{minipage}[t]{.4\textwidth}
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\isa{%
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*}
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(*<*)lemma "P" proof-(*>*)
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show "P"
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proof (rule ccontr)
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assume "\<not>P"
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txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
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show "False" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
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qed(*<*)oops(*>*)
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text_raw {* }
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\end{minipage}
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\end{tabular}
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\medskip
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\begin{isamarkuptext}%
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The name @{thm[source] ccontr} stands for ``classical contradiction''.
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How to prove quantified formulas:
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\end{isamarkuptext}%
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\begin{tabular}{@ {}ll@ {}}
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\begin{minipage}[t]{.4\textwidth}
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\isa{%
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*}
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(*<*)lemma "ALL x. P x" proof-(*>*)
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show "\<forall>x. P(x)"
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proof
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fix x
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txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
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show "P(x)" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
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qed(*<*)oops(*>*)
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text_raw {* }
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\end{minipage}
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&
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\begin{minipage}[t]{.4\textwidth}
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\isa{%
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*}
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(*<*)lemma "EX x. P(x)" proof-(*>*)
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show "\<exists>x. P(x)"
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proof
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txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
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show "P(witness)" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
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qed
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(*<*)oops(*>*)
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text_raw {* }
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\end{minipage}
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\end{tabular}
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\medskip
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\begin{isamarkuptext}%
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In the proof of \noquotes{@{prop[source]"\<forall>x. P(x)"}},
|
47704
|
362 |
the step \isacom{fix}~@{text x} introduces a locally fixed variable @{text x}
|
47269
|
363 |
into the subproof, the proverbial ``arbitrary but fixed value''.
|
|
364 |
Instead of @{text x} we could have chosen any name in the subproof.
|
|
365 |
In the proof of \noquotes{@{prop[source]"\<exists>x. P(x)"}},
|
|
366 |
@{text witness} is some arbitrary
|
|
367 |
term for which we can prove that it satisfies @{text P}.
|
|
368 |
|
|
369 |
How to reason forward from \noquotes{@{prop[source] "\<exists>x. P(x)"}}:
|
|
370 |
\end{isamarkuptext}%
|
|
371 |
*}
|
|
372 |
(*<*)lemma True proof- assume 1: "EX x. P x"(*>*)
|
|
373 |
have "\<exists>x. P(x)" (*<*)by(rule 1)(*>*)txt_raw{*\ $\dots$\\*}
|
|
374 |
then obtain x where p: "P(x)" by blast
|
|
375 |
(*<*)oops(*>*)
|
|
376 |
text{*
|
|
377 |
After the \isacom{obtain} step, @{text x} (we could have chosen any name)
|
|
378 |
is a fixed local
|
|
379 |
variable, and @{text p} is the name of the fact
|
|
380 |
\noquotes{@{prop[source] "P(x)"}}.
|
|
381 |
This pattern works for one or more @{text x}.
|
|
382 |
As an example of the \isacom{obtain} command, here is the proof of
|
|
383 |
Cantor's theorem in more detail:
|
|
384 |
*}
|
|
385 |
|
|
386 |
lemma "\<not> surj(f :: 'a \<Rightarrow> 'a set)"
|
|
387 |
proof
|
|
388 |
assume "surj f"
|
|
389 |
hence "\<exists>a. {x. x \<notin> f x} = f a" by(auto simp: surj_def)
|
|
390 |
then obtain a where "{x. x \<notin> f x} = f a" by blast
|
|
391 |
hence "a \<notin> f a \<longleftrightarrow> a \<in> f a" by blast
|
|
392 |
thus "False" by blast
|
|
393 |
qed
|
|
394 |
|
|
395 |
text_raw{*
|
|
396 |
\begin{isamarkuptext}%
|
47306
|
397 |
|
|
398 |
Finally, how to prove set equality and subset relationship:
|
47269
|
399 |
\end{isamarkuptext}%
|
|
400 |
\begin{tabular}{@ {}ll@ {}}
|
|
401 |
\begin{minipage}[t]{.4\textwidth}
|
|
402 |
\isa{%
|
|
403 |
*}
|
|
404 |
(*<*)lemma "A = (B::'a set)" proof-(*>*)
|
|
405 |
show "A = B"
|
|
406 |
proof
|
|
407 |
show "A \<subseteq> B" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
|
|
408 |
next
|
|
409 |
show "B \<subseteq> A" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
|
|
410 |
qed(*<*)qed(*>*)
|
|
411 |
|
|
412 |
text_raw {* }
|
|
413 |
\end{minipage}
|
|
414 |
&
|
|
415 |
\begin{minipage}[t]{.4\textwidth}
|
|
416 |
\isa{%
|
|
417 |
*}
|
|
418 |
(*<*)lemma "A <= (B::'a set)" proof-(*>*)
|
|
419 |
show "A \<subseteq> B"
|
|
420 |
proof
|
|
421 |
fix x
|
|
422 |
assume "x \<in> A"
|
|
423 |
txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
|
|
424 |
show "x \<in> B" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
|
|
425 |
qed(*<*)qed(*>*)
|
|
426 |
|
|
427 |
text_raw {* }
|
|
428 |
\end{minipage}
|
|
429 |
\end{tabular}
|
|
430 |
\begin{isamarkuptext}%
|
|
431 |
\section{Streamlining proofs}
|
|
432 |
|
|
433 |
\subsection{Pattern matching and quotations}
|
|
434 |
|
|
435 |
In the proof patterns shown above, formulas are often duplicated.
|
|
436 |
This can make the text harder to read, write and maintain. Pattern matching
|
|
437 |
is an abbreviation mechanism to avoid such duplication. Writing
|
|
438 |
\begin{quote}
|
|
439 |
\isacom{show} \ \textit{formula} @{text"("}\isacom{is} \textit{pattern}@{text")"}
|
|
440 |
\end{quote}
|
|
441 |
matches the pattern against the formula, thus instantiating the unknowns in
|
|
442 |
the pattern for later use. As an example, consider the proof pattern for
|
|
443 |
@{text"\<longleftrightarrow>"}:
|
|
444 |
\end{isamarkuptext}%
|
|
445 |
*}
|
|
446 |
(*<*)lemma "formula\<^isub>1 \<longleftrightarrow> formula\<^isub>2" proof-(*>*)
|
|
447 |
show "formula\<^isub>1 \<longleftrightarrow> formula\<^isub>2" (is "?L \<longleftrightarrow> ?R")
|
|
448 |
proof
|
|
449 |
assume "?L"
|
|
450 |
txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
|
|
451 |
show "?R" (*<*)sorry(*>*) txt_raw{*\ $\dots$\\*}
|
|
452 |
next
|
|
453 |
assume "?R"
|
|
454 |
txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
|
|
455 |
show "?L" (*<*)sorry(*>*) txt_raw{*\ $\dots$\\*}
|
|
456 |
qed(*<*)qed(*>*)
|
|
457 |
|
|
458 |
text{* Instead of duplicating @{text"formula\<^isub>i"} in the text, we introduce
|
|
459 |
the two abbreviations @{text"?L"} and @{text"?R"} by pattern matching.
|
|
460 |
Pattern matching works wherever a formula is stated, in particular
|
|
461 |
with \isacom{have} and \isacom{lemma}.
|
|
462 |
|
|
463 |
The unknown @{text"?thesis"} is implicitly matched against any goal stated by
|
|
464 |
\isacom{lemma} or \isacom{show}. Here is a typical example: *}
|
|
465 |
|
|
466 |
lemma "formula"
|
|
467 |
proof -
|
|
468 |
txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
|
|
469 |
show ?thesis (*<*)sorry(*>*) txt_raw{*\ $\dots$\\*}
|
|
470 |
qed
|
|
471 |
|
|
472 |
text{*
|
|
473 |
Unknowns can also be instantiated with \isacom{let} commands
|
|
474 |
\begin{quote}
|
|
475 |
\isacom{let} @{text"?t"} = @{text"\""}\textit{some-big-term}@{text"\""}
|
|
476 |
\end{quote}
|
|
477 |
Later proof steps can refer to @{text"?t"}:
|
|
478 |
\begin{quote}
|
|
479 |
\isacom{have} @{text"\""}\dots @{text"?t"} \dots@{text"\""}
|
|
480 |
\end{quote}
|
|
481 |
\begin{warn}
|
|
482 |
Names of facts are introduced with @{text"name:"} and refer to proved
|
|
483 |
theorems. Unknowns @{text"?X"} refer to terms or formulas.
|
|
484 |
\end{warn}
|
|
485 |
|
|
486 |
Although abbreviations shorten the text, the reader needs to remember what
|
|
487 |
they stand for. Similarly for names of facts. Names like @{text 1}, @{text 2}
|
|
488 |
and @{text 3} are not helpful and should only be used in short proofs. For
|
47704
|
489 |
longer proofs, descriptive names are better. But look at this example:
|
47269
|
490 |
\begin{quote}
|
|
491 |
\isacom{have} \ @{text"x_gr_0: \"x > 0\""}\\
|
|
492 |
$\vdots$\\
|
|
493 |
\isacom{from} @{text "x_gr_0"} \dots
|
|
494 |
\end{quote}
|
|
495 |
The name is longer than the fact it stands for! Short facts do not need names,
|
|
496 |
one can refer to them easily by quoting them:
|
|
497 |
\begin{quote}
|
|
498 |
\isacom{have} \ @{text"\"x > 0\""}\\
|
|
499 |
$\vdots$\\
|
|
500 |
\isacom{from} @{text "`x>0`"} \dots
|
|
501 |
\end{quote}
|
|
502 |
Note that the quotes around @{text"x>0"} are \concept{back quotes}.
|
|
503 |
They refer to the fact not by name but by value.
|
|
504 |
|
|
505 |
\subsection{\isacom{moreover}}
|
|
506 |
|
|
507 |
Sometimes one needs a number of facts to enable some deduction. Of course
|
|
508 |
one can name these facts individually, as shown on the right,
|
|
509 |
but one can also combine them with \isacom{moreover}, as shown on the left:
|
|
510 |
*}text_raw{*
|
|
511 |
\begin{tabular}{@ {}ll@ {}}
|
|
512 |
\begin{minipage}[t]{.4\textwidth}
|
|
513 |
\isa{%
|
|
514 |
*}
|
|
515 |
(*<*)lemma "P" proof-(*>*)
|
|
516 |
have "P\<^isub>1" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
|
|
517 |
moreover have "P\<^isub>2" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
|
|
518 |
moreover
|
|
519 |
txt_raw{*\\$\vdots$\\\hspace{-1.4ex}*}(*<*)have "True" ..(*>*)
|
|
520 |
moreover have "P\<^isub>n" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
|
|
521 |
ultimately have "P" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
|
|
522 |
(*<*)oops(*>*)
|
|
523 |
|
|
524 |
text_raw {* }
|
|
525 |
\end{minipage}
|
|
526 |
&
|
|
527 |
\qquad
|
|
528 |
\begin{minipage}[t]{.4\textwidth}
|
|
529 |
\isa{%
|
|
530 |
*}
|
|
531 |
(*<*)lemma "P" proof-(*>*)
|
|
532 |
have lab\<^isub>1: "P\<^isub>1" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
|
|
533 |
have lab\<^isub>2: "P\<^isub>2" (*<*)sorry(*>*)txt_raw{*\ $\dots$*}
|
|
534 |
txt_raw{*\\$\vdots$\\\hspace{-1.4ex}*}
|
|
535 |
have lab\<^isub>n: "P\<^isub>n" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
|
|
536 |
from lab\<^isub>1 lab\<^isub>2 txt_raw{*\ $\dots$\\*}
|
|
537 |
have "P" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
|
|
538 |
(*<*)oops(*>*)
|
|
539 |
|
|
540 |
text_raw {* }
|
|
541 |
\end{minipage}
|
|
542 |
\end{tabular}
|
|
543 |
\begin{isamarkuptext}%
|
|
544 |
The \isacom{moreover} version is no shorter but expresses the structure more
|
|
545 |
clearly and avoids new names.
|
|
546 |
|
|
547 |
\subsection{Raw proof blocks}
|
|
548 |
|
47306
|
549 |
Sometimes one would like to prove some lemma locally within a proof.
|
47269
|
550 |
A lemma that shares the current context of assumptions but that
|
|
551 |
has its own assumptions and is generalised over its locally fixed
|
|
552 |
variables at the end. This is what a \concept{raw proof block} does:
|
|
553 |
\begin{quote}
|
|
554 |
@{text"{"} \isacom{fix} @{text"x\<^isub>1 \<dots> x\<^isub>n"}\\
|
|
555 |
\mbox{}\ \ \ \isacom{assume} @{text"A\<^isub>1 \<dots> A\<^isub>m"}\\
|
|
556 |
\mbox{}\ \ \ $\vdots$\\
|
|
557 |
\mbox{}\ \ \ \isacom{have} @{text"B"}\\
|
|
558 |
@{text"}"}
|
|
559 |
\end{quote}
|
|
560 |
proves @{text"\<lbrakk> A\<^isub>1; \<dots> ; A\<^isub>m \<rbrakk> \<Longrightarrow> B"}
|
|
561 |
where all @{text"x\<^isub>i"} have been replaced by unknowns @{text"?x\<^isub>i"}.
|
|
562 |
\begin{warn}
|
|
563 |
The conclusion of a raw proof block is \emph{not} indicated by \isacom{show}
|
|
564 |
but is simply the final \isacom{have}.
|
|
565 |
\end{warn}
|
|
566 |
|
|
567 |
As an example we prove a simple fact about divisibility on integers.
|
|
568 |
The definition of @{text "dvd"} is @{thm dvd_def}.
|
|
569 |
\end{isamarkuptext}%
|
|
570 |
*}
|
|
571 |
|
|
572 |
lemma fixes a b :: int assumes "b dvd (a+b)" shows "b dvd a"
|
|
573 |
proof -
|
|
574 |
{ fix k assume k: "a+b = b*k"
|
|
575 |
have "\<exists>k'. a = b*k'"
|
|
576 |
proof
|
|
577 |
show "a = b*(k - 1)" using k by(simp add: algebra_simps)
|
|
578 |
qed }
|
|
579 |
then show ?thesis using assms by(auto simp add: dvd_def)
|
|
580 |
qed
|
|
581 |
|
|
582 |
text{* Note that the result of a raw proof block has no name. In this example
|
|
583 |
it was directly piped (via \isacom{then}) into the final proof, but it can
|
|
584 |
also be named for later reference: you simply follow the block directly by a
|
|
585 |
\isacom{note} command:
|
|
586 |
\begin{quote}
|
|
587 |
\isacom{note} \ @{text"name = this"}
|
|
588 |
\end{quote}
|
|
589 |
This introduces a new name @{text name} that refers to @{text this},
|
|
590 |
the fact just proved, in this case the preceding block. In general,
|
|
591 |
\isacom{note} introduces a new name for one or more facts.
|
|
592 |
|
|
593 |
\section{Case distinction and induction}
|
|
594 |
|
|
595 |
\subsection{Datatype case distinction}
|
|
596 |
|
|
597 |
We have seen case distinction on formulas. Now we want to distinguish
|
|
598 |
which form some term takes: is it @{text 0} or of the form @{term"Suc n"},
|
|
599 |
is it @{term"[]"} or of the form @{term"x#xs"}, etc. Here is a typical example
|
|
600 |
proof by case distinction on the form of @{text xs}:
|
|
601 |
*}
|
|
602 |
|
|
603 |
lemma "length(tl xs) = length xs - 1"
|
|
604 |
proof (cases xs)
|
|
605 |
assume "xs = []"
|
|
606 |
thus ?thesis by simp
|
|
607 |
next
|
|
608 |
fix y ys assume "xs = y#ys"
|
|
609 |
thus ?thesis by simp
|
|
610 |
qed
|
|
611 |
|
|
612 |
text{* Function @{text tl} (''tail'') is defined by @{thm tl.simps(1)} and
|
|
613 |
@{thm tl.simps(2)}. Note that the result type of @{const length} is @{typ nat}
|
|
614 |
and @{prop"0 - 1 = (0::nat)"}.
|
|
615 |
|
|
616 |
This proof pattern works for any term @{text t} whose type is a datatype.
|
|
617 |
The goal has to be proved for each constructor @{text C}:
|
|
618 |
\begin{quote}
|
|
619 |
\isacom{fix} \ @{text"x\<^isub>1 \<dots> x\<^isub>n"} \isacom{assume} @{text"\"t = C x\<^isub>1 \<dots> x\<^isub>n\""}
|
|
620 |
\end{quote}
|
|
621 |
Each case can be written in a more compact form by means of the \isacom{case}
|
|
622 |
command:
|
|
623 |
\begin{quote}
|
|
624 |
\isacom{case} @{text "(C x\<^isub>1 \<dots> x\<^isub>n)"}
|
|
625 |
\end{quote}
|
47704
|
626 |
This is equivalent to the explicit \isacom{fix}-\isacom{assume} line
|
47269
|
627 |
but also gives the assumption @{text"\"t = C x\<^isub>1 \<dots> x\<^isub>n\""} a name: @{text C},
|
|
628 |
like the constructor.
|
|
629 |
Here is the \isacom{case} version of the proof above:
|
|
630 |
*}
|
|
631 |
(*<*)lemma "length(tl xs) = length xs - 1"(*>*)
|
|
632 |
proof (cases xs)
|
|
633 |
case Nil
|
|
634 |
thus ?thesis by simp
|
|
635 |
next
|
|
636 |
case (Cons y ys)
|
|
637 |
thus ?thesis by simp
|
|
638 |
qed
|
|
639 |
|
|
640 |
text{* Remember that @{text Nil} and @{text Cons} are the alphanumeric names
|
|
641 |
for @{text"[]"} and @{text"#"}. The names of the assumptions
|
|
642 |
are not used because they are directly piped (via \isacom{thus})
|
|
643 |
into the proof of the claim.
|
|
644 |
|
|
645 |
\subsection{Structural induction}
|
|
646 |
|
|
647 |
We illustrate structural induction with an example based on natural numbers:
|
|
648 |
the sum (@{text"\<Sum>"}) of the first @{text n} natural numbers
|
|
649 |
(@{text"{0..n::nat}"}) is equal to \mbox{@{term"n*(n+1) div 2::nat"}}.
|
|
650 |
Never mind the details, just focus on the pattern:
|
|
651 |
*}
|
|
652 |
|
|
653 |
lemma "\<Sum>{0..n::nat} = n*(n+1) div 2" (is "?P n")
|
|
654 |
proof (induction n)
|
|
655 |
show "\<Sum>{0..0::nat} = 0*(0+1) div 2" by simp
|
|
656 |
next
|
|
657 |
fix n assume "\<Sum>{0..n::nat} = n*(n+1) div 2"
|
|
658 |
thus "\<Sum>{0..Suc n::nat} = Suc n*(Suc n+1) div 2" by simp
|
|
659 |
qed
|
|
660 |
|
|
661 |
text{* Except for the rewrite steps, everything is explicitly given. This
|
|
662 |
makes the proof easily readable, but the duplication means it is tedious to
|
|
663 |
write and maintain. Here is how pattern
|
|
664 |
matching can completely avoid any duplication: *}
|
|
665 |
|
|
666 |
lemma "\<Sum>{0..n::nat} = n*(n+1) div 2" (is "?P n")
|
|
667 |
proof (induction n)
|
|
668 |
show "?P 0" by simp
|
|
669 |
next
|
|
670 |
fix n assume "?P n"
|
|
671 |
thus "?P(Suc n)" by simp
|
|
672 |
qed
|
|
673 |
|
|
674 |
text{* The first line introduces an abbreviation @{text"?P n"} for the goal.
|
|
675 |
Pattern matching @{text"?P n"} with the goal instantiates @{text"?P"} to the
|
|
676 |
function @{term"\<lambda>n. \<Sum>{0..n::nat} = n*(n+1) div 2"}. Now the proposition to
|
|
677 |
be proved in the base case can be written as @{text"?P 0"}, the induction
|
|
678 |
hypothesis as @{text"?P n"}, and the conclusion of the induction step as
|
|
679 |
@{text"?P(Suc n)"}.
|
|
680 |
|
|
681 |
Induction also provides the \isacom{case} idiom that abbreviates
|
|
682 |
the \isacom{fix}-\isacom{assume} step. The above proof becomes
|
|
683 |
*}
|
|
684 |
(*<*)lemma "\<Sum>{0..n::nat} = n*(n+1) div 2"(*>*)
|
|
685 |
proof (induction n)
|
|
686 |
case 0
|
|
687 |
show ?case by simp
|
|
688 |
next
|
|
689 |
case (Suc n)
|
|
690 |
thus ?case by simp
|
|
691 |
qed
|
|
692 |
|
|
693 |
text{*
|
|
694 |
The unknown @{text "?case"} is set in each case to the required
|
|
695 |
claim, i.e.\ @{text"?P 0"} and \mbox{@{text"?P(Suc n)"}} in the above proof,
|
|
696 |
without requiring the user to define a @{text "?P"}. The general
|
|
697 |
pattern for induction over @{typ nat} is shown on the left-hand side:
|
|
698 |
*}text_raw{*
|
|
699 |
\begin{tabular}{@ {}ll@ {}}
|
|
700 |
\begin{minipage}[t]{.4\textwidth}
|
|
701 |
\isa{%
|
|
702 |
*}
|
|
703 |
(*<*)lemma "P(n::nat)" proof -(*>*)
|
|
704 |
show "P(n)"
|
|
705 |
proof (induction n)
|
|
706 |
case 0
|
|
707 |
txt_raw{*\\\mbox{}\ \ $\vdots$\\\mbox{}\hspace{-1ex}*}
|
|
708 |
show ?case (*<*)sorry(*>*) txt_raw{*\ $\dots$\\*}
|
|
709 |
next
|
|
710 |
case (Suc n)
|
|
711 |
txt_raw{*\\\mbox{}\ \ $\vdots$\\\mbox{}\hspace{-1ex}*}
|
|
712 |
show ?case (*<*)sorry(*>*) txt_raw{*\ $\dots$\\*}
|
|
713 |
qed(*<*)qed(*>*)
|
|
714 |
|
|
715 |
text_raw {* }
|
|
716 |
\end{minipage}
|
|
717 |
&
|
|
718 |
\begin{minipage}[t]{.4\textwidth}
|
|
719 |
~\\
|
|
720 |
~\\
|
|
721 |
\isacom{let} @{text"?case = \"P(0)\""}\\
|
|
722 |
~\\
|
|
723 |
~\\
|
|
724 |
~\\[1ex]
|
|
725 |
\isacom{fix} @{text n} \isacom{assume} @{text"Suc: \"P(n)\""}\\
|
|
726 |
\isacom{let} @{text"?case = \"P(Suc n)\""}\\
|
|
727 |
\end{minipage}
|
|
728 |
\end{tabular}
|
|
729 |
\medskip
|
|
730 |
*}
|
|
731 |
text{*
|
|
732 |
On the right side you can see what the \isacom{case} command
|
|
733 |
on the left stands for.
|
|
734 |
|
|
735 |
In case the goal is an implication, induction does one more thing: the
|
|
736 |
proposition to be proved in each case is not the whole implication but only
|
|
737 |
its conclusion; the premises of the implication are immediately made
|
|
738 |
assumptions of that case. That is, if in the above proof we replace
|
|
739 |
\isacom{show}~@{text"P(n)"} by
|
|
740 |
\mbox{\isacom{show}~@{text"A(n) \<Longrightarrow> P(n)"}}
|
|
741 |
then \isacom{case}~@{text 0} stands for
|
|
742 |
\begin{quote}
|
|
743 |
\isacom{assume} \ @{text"0: \"A(0)\""}\\
|
|
744 |
\isacom{let} @{text"?case = \"P(0)\""}
|
|
745 |
\end{quote}
|
|
746 |
and \isacom{case}~@{text"(Suc n)"} stands for
|
|
747 |
\begin{quote}
|
|
748 |
\isacom{fix} @{text n}\\
|
|
749 |
\isacom{assume} @{text"Suc:"}
|
47306
|
750 |
\begin{tabular}[t]{l}@{text"\"A(n) \<Longrightarrow> P(n)\""}\\@{text"\"A(Suc n)\""}\end{tabular}\\
|
47269
|
751 |
\isacom{let} @{text"?case = \"P(Suc n)\""}
|
|
752 |
\end{quote}
|
|
753 |
The list of assumptions @{text Suc} is actually subdivided
|
|
754 |
into @{text"Suc.IH"}, the induction hypotheses (here @{text"A(n) \<Longrightarrow> P(n)"})
|
|
755 |
and @{text"Suc.prems"}, the premises of the goal being proved
|
|
756 |
(here @{text"A(Suc n)"}).
|
|
757 |
|
|
758 |
Induction works for any datatype.
|
|
759 |
Proving a goal @{text"\<lbrakk> A\<^isub>1(x); \<dots>; A\<^isub>k(x) \<rbrakk> \<Longrightarrow> P(x)"}
|
|
760 |
by induction on @{text x} generates a proof obligation for each constructor
|
|
761 |
@{text C} of the datatype. The command @{text"case (C x\<^isub>1 \<dots> x\<^isub>n)"}
|
|
762 |
performs the following steps:
|
|
763 |
\begin{enumerate}
|
|
764 |
\item \isacom{fix} @{text"x\<^isub>1 \<dots> x\<^isub>n"}
|
|
765 |
\item \isacom{assume} the induction hypotheses (calling them @{text C.IH})
|
|
766 |
and the premises \mbox{@{text"A\<^isub>i(C x\<^isub>1 \<dots> x\<^isub>n)"}} (calling them @{text"C.prems"})
|
|
767 |
and calling the whole list @{text C}
|
|
768 |
\item \isacom{let} @{text"?case = \"P(C x\<^isub>1 \<dots> x\<^isub>n)\""}
|
|
769 |
\end{enumerate}
|
|
770 |
|
|
771 |
\subsection{Rule induction}
|
|
772 |
|
|
773 |
Recall the inductive and recursive definitions of even numbers in
|
|
774 |
\autoref{sec:inductive-defs}:
|
|
775 |
*}
|
|
776 |
|
|
777 |
inductive ev :: "nat \<Rightarrow> bool" where
|
|
778 |
ev0: "ev 0" |
|
|
779 |
evSS: "ev n \<Longrightarrow> ev(Suc(Suc n))"
|
|
780 |
|
|
781 |
fun even :: "nat \<Rightarrow> bool" where
|
|
782 |
"even 0 = True" |
|
|
783 |
"even (Suc 0) = False" |
|
|
784 |
"even (Suc(Suc n)) = even n"
|
|
785 |
|
|
786 |
text{* We recast the proof of @{prop"ev n \<Longrightarrow> even n"} in Isar. The
|
|
787 |
left column shows the actual proof text, the right column shows
|
|
788 |
the implicit effect of the two \isacom{case} commands:*}text_raw{*
|
|
789 |
\begin{tabular}{@ {}l@ {\qquad}l@ {}}
|
|
790 |
\begin{minipage}[t]{.5\textwidth}
|
|
791 |
\isa{%
|
|
792 |
*}
|
|
793 |
|
|
794 |
lemma "ev n \<Longrightarrow> even n"
|
|
795 |
proof(induction rule: ev.induct)
|
|
796 |
case ev0
|
|
797 |
show ?case by simp
|
|
798 |
next
|
|
799 |
case evSS
|
|
800 |
|
|
801 |
|
|
802 |
|
|
803 |
thus ?case by simp
|
|
804 |
qed
|
|
805 |
|
|
806 |
text_raw {* }
|
|
807 |
\end{minipage}
|
|
808 |
&
|
|
809 |
\begin{minipage}[t]{.5\textwidth}
|
|
810 |
~\\
|
|
811 |
~\\
|
|
812 |
\isacom{let} @{text"?case = \"even 0\""}\\
|
|
813 |
~\\
|
|
814 |
~\\
|
|
815 |
\isacom{fix} @{text n}\\
|
|
816 |
\isacom{assume} @{text"evSS:"}
|
47306
|
817 |
\begin{tabular}[t]{l} @{text"\"ev n\""}\\@{text"\"even n\""}\end{tabular}\\
|
|
818 |
\isacom{let} @{text"?case = \"even(Suc(Suc n))\""}\\
|
47269
|
819 |
\end{minipage}
|
|
820 |
\end{tabular}
|
|
821 |
\medskip
|
|
822 |
*}
|
|
823 |
text{*
|
|
824 |
The proof resembles structural induction, but the induction rule is given
|
|
825 |
explicitly and the names of the cases are the names of the rules in the
|
|
826 |
inductive definition.
|
|
827 |
Let us examine the two assumptions named @{thm[source]evSS}:
|
|
828 |
@{prop "ev n"} is the premise of rule @{thm[source]evSS}, which we may assume
|
|
829 |
because we are in the case where that rule was used; @{prop"even n"}
|
|
830 |
is the induction hypothesis.
|
|
831 |
\begin{warn}
|
|
832 |
Because each \isacom{case} command introduces a list of assumptions
|
|
833 |
named like the case name, which is the name of a rule of the inductive
|
|
834 |
definition, those rules now need to be accessed with a qualified name, here
|
|
835 |
@{thm[source] ev.ev0} and @{thm[source] ev.evSS}
|
|
836 |
\end{warn}
|
|
837 |
|
|
838 |
In the case @{thm[source]evSS} of the proof above we have pretended that the
|
|
839 |
system fixes a variable @{text n}. But unless the user provides the name
|
|
840 |
@{text n}, the system will just invent its own name that cannot be referred
|
|
841 |
to. In the above proof, we do not need to refer to it, hence we do not give
|
|
842 |
it a specific name. In case one needs to refer to it one writes
|
|
843 |
\begin{quote}
|
|
844 |
\isacom{case} @{text"(evSS m)"}
|
|
845 |
\end{quote}
|
|
846 |
just like \isacom{case}~@{text"(Suc n)"} in earlier structural inductions.
|
|
847 |
The name @{text m} is an arbitrary choice. As a result,
|
|
848 |
case @{thm[source] evSS} is derived from a renamed version of
|
|
849 |
rule @{thm[source] evSS}: @{text"ev m \<Longrightarrow> ev(Suc(Suc m))"}.
|
|
850 |
Here is an example with a (contrived) intermediate step that refers to @{text m}:
|
|
851 |
*}
|
|
852 |
|
|
853 |
lemma "ev n \<Longrightarrow> even n"
|
|
854 |
proof(induction rule: ev.induct)
|
|
855 |
case ev0 show ?case by simp
|
|
856 |
next
|
|
857 |
case (evSS m)
|
|
858 |
have "even(Suc(Suc m)) = even m" by simp
|
|
859 |
thus ?case using `even m` by blast
|
|
860 |
qed
|
|
861 |
|
|
862 |
text{*
|
|
863 |
\indent
|
|
864 |
In general, let @{text I} be a (for simplicity unary) inductively defined
|
|
865 |
predicate and let the rules in the definition of @{text I}
|
|
866 |
be called @{text "rule\<^isub>1"}, \dots, @{text "rule\<^isub>n"}. A proof by rule
|
|
867 |
induction follows this pattern:
|
|
868 |
*}
|
|
869 |
|
|
870 |
(*<*)
|
|
871 |
inductive I where rule\<^isub>1: "I()" | rule\<^isub>2: "I()" | rule\<^isub>n: "I()"
|
|
872 |
lemma "I x \<Longrightarrow> P x" proof-(*>*)
|
|
873 |
show "I x \<Longrightarrow> P x"
|
|
874 |
proof(induction rule: I.induct)
|
|
875 |
case rule\<^isub>1
|
|
876 |
txt_raw{*\\[-.4ex]\mbox{}\ \ $\vdots$\\[-.4ex]\mbox{}\hspace{-1ex}*}
|
|
877 |
show ?case (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
|
|
878 |
next
|
|
879 |
txt_raw{*\\[-.4ex]$\vdots$\\[-.4ex]\mbox{}\hspace{-1ex}*}
|
|
880 |
(*<*)
|
|
881 |
case rule\<^isub>2
|
|
882 |
show ?case sorry
|
|
883 |
(*>*)
|
|
884 |
next
|
|
885 |
case rule\<^isub>n
|
|
886 |
txt_raw{*\\[-.4ex]\mbox{}\ \ $\vdots$\\[-.4ex]\mbox{}\hspace{-1ex}*}
|
|
887 |
show ?case (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
|
|
888 |
qed(*<*)qed(*>*)
|
|
889 |
|
|
890 |
text{*
|
|
891 |
One can provide explicit variable names by writing
|
|
892 |
\isacom{case}~@{text"(rule\<^isub>i x\<^isub>1 \<dots> x\<^isub>k)"}, thus renaming the first @{text k}
|
|
893 |
free variables in rule @{text i} to @{text"x\<^isub>1 \<dots> x\<^isub>k"},
|
|
894 |
going through rule @{text i} from left to right.
|
|
895 |
|
|
896 |
\subsection{Assumption naming}
|
|
897 |
|
|
898 |
In any induction, \isacom{case}~@{text name} sets up a list of assumptions
|
|
899 |
also called @{text name}, which is subdivided into three parts:
|
|
900 |
\begin{description}
|
|
901 |
\item[@{text name.IH}] contains the induction hypotheses.
|
|
902 |
\item[@{text name.hyps}] contains all the other hypotheses of this case in the
|
|
903 |
induction rule. For rule inductions these are the hypotheses of rule
|
|
904 |
@{text name}, for structural inductions these are empty.
|
|
905 |
\item[@{text name.prems}] contains the (suitably instantiated) premises
|
|
906 |
of the statement being proved, i.e. the @{text A\<^isub>i} when
|
|
907 |
proving @{text"\<lbrakk> A\<^isub>1; \<dots>; A\<^isub>n \<rbrakk> \<Longrightarrow> A"}.
|
|
908 |
\end{description}
|
|
909 |
\begin{warn}
|
|
910 |
Proof method @{text induct} differs from @{text induction}
|
|
911 |
only in this naming policy: @{text induct} does not distinguish
|
|
912 |
@{text IH} from @{text hyps} but subsumes @{text IH} under @{text hyps}.
|
|
913 |
\end{warn}
|
|
914 |
|
|
915 |
More complicated inductive proofs than the ones we have seen so far
|
|
916 |
often need to refer to specific assumptions---just @{text name} or even
|
|
917 |
@{text name.prems} and @{text name.IH} can be too unspecific.
|
|
918 |
This is where the indexing of fact lists comes in handy, e.g.\
|
|
919 |
@{text"name.IH(2)"} or @{text"name.prems(1-2)"}.
|
|
920 |
|
|
921 |
\subsection{Rule inversion}
|
|
922 |
|
|
923 |
Rule inversion is case distinction on which rule could have been used to
|
|
924 |
derive some fact. The name \concept{rule inversion} emphasizes that we are
|
|
925 |
reasoning backwards: by which rules could some given fact have been proved?
|
|
926 |
For the inductive definition of @{const ev}, rule inversion can be summarized
|
|
927 |
like this:
|
|
928 |
@{prop[display]"ev n \<Longrightarrow> n = 0 \<or> (EX k. n = Suc(Suc k) \<and> ev k)"}
|
|
929 |
The realisation in Isabelle is a case distinction.
|
|
930 |
A simple example is the proof that @{prop"ev n \<Longrightarrow> ev (n - 2)"}. We
|
|
931 |
already went through the details informally in \autoref{sec:Logic:even}. This
|
|
932 |
is the Isar proof:
|
|
933 |
*}
|
|
934 |
(*<*)
|
|
935 |
notepad
|
|
936 |
begin fix n
|
|
937 |
(*>*)
|
|
938 |
assume "ev n"
|
|
939 |
from this have "ev(n - 2)"
|
|
940 |
proof cases
|
|
941 |
case ev0 thus "ev(n - 2)" by (simp add: ev.ev0)
|
|
942 |
next
|
|
943 |
case (evSS k) thus "ev(n - 2)" by (simp add: ev.evSS)
|
|
944 |
qed
|
|
945 |
(*<*)
|
|
946 |
end
|
|
947 |
(*>*)
|
|
948 |
|
|
949 |
text{* The key point here is that a case distinction over some inductively
|
|
950 |
defined predicate is triggered by piping the given fact
|
|
951 |
(here: \isacom{from}~@{text this}) into a proof by @{text cases}.
|
|
952 |
Let us examine the assumptions available in each case. In case @{text ev0}
|
|
953 |
we have @{text"n = 0"} and in case @{text evSS} we have @{prop"n = Suc(Suc k)"}
|
|
954 |
and @{prop"ev k"}. In each case the assumptions are available under the name
|
|
955 |
of the case; there is no fine grained naming schema like for induction.
|
|
956 |
|
47704
|
957 |
Sometimes some rules could not have been used to derive the given fact
|
47269
|
958 |
because constructors clash. As an extreme example consider
|
|
959 |
rule inversion applied to @{prop"ev(Suc 0)"}: neither rule @{text ev0} nor
|
|
960 |
rule @{text evSS} can yield @{prop"ev(Suc 0)"} because @{text"Suc 0"} unifies
|
|
961 |
neither with @{text 0} nor with @{term"Suc(Suc n)"}. Impossible cases do not
|
|
962 |
have to be proved. Hence we can prove anything from @{prop"ev(Suc 0)"}:
|
|
963 |
*}
|
|
964 |
(*<*)
|
|
965 |
notepad begin fix P
|
|
966 |
(*>*)
|
|
967 |
assume "ev(Suc 0)" then have P by cases
|
|
968 |
(*<*)
|
|
969 |
end
|
|
970 |
(*>*)
|
|
971 |
|
|
972 |
text{* That is, @{prop"ev(Suc 0)"} is simply not provable: *}
|
|
973 |
|
|
974 |
lemma "\<not> ev(Suc 0)"
|
|
975 |
proof
|
|
976 |
assume "ev(Suc 0)" then show False by cases
|
|
977 |
qed
|
|
978 |
|
|
979 |
text{* Normally not all cases will be impossible. As a simple exercise,
|
|
980 |
prove that \mbox{@{prop"\<not> ev(Suc(Suc(Suc 0)))"}.}
|
|
981 |
*}
|
|
982 |
|
|
983 |
(*
|
|
984 |
lemma "\<not> ev(Suc(Suc(Suc 0)))"
|
|
985 |
proof
|
|
986 |
assume "ev(Suc(Suc(Suc 0)))"
|
|
987 |
then show False
|
|
988 |
proof cases
|
|
989 |
case evSS
|
|
990 |
from `ev(Suc 0)` show False by cases
|
|
991 |
qed
|
|
992 |
qed
|
|
993 |
*)
|
|
994 |
|
|
995 |
(*<*)
|
|
996 |
end
|
|
997 |
(*>*)
|