676 lines
18 KiB
TeX
676 lines
18 KiB
TeX
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\part{Basic Tactics}
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\frame[plain]{\partpage}
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\begin{frame}
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\frametitle{Syntax of Tactics in HOL}
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\begin{itemize}
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\item originally tactics were written all in capital letters with underscores\\
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Example: \hol{ALL\_TAC}
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\item since 2010 more and more tactics have overloaded lower-case syntax\\
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Example: \hol{all\_tac}
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\item sometimes, the lower-case version is shortened\\
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Example: \hol{REPEAT}, \hol{rpt}
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\item sometimes, there is special syntax\\
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Example: \hol{THEN}, \hol{\textbsl{}\textbsl{}}, \hol{>>}
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\item which one to use is mostly a matter of personal taste
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\begin{itemize}
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\item all-capital names are hard to read and type
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\item however, not for all tactics there are lower-case versions
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\item mixed lower- and upper-case tactics are even harder to read
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\item often shortened lower-case name is not \textit{speaking}
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\end{itemize}
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\end{itemize}
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\bottomstatement{In the lecture we will use mostly the old-style names.}
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\end{frame}
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\section{Basic Tactics}
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\begin{frame}
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\frametitle{Some Basic Tactics}
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\begin{tabular}{ll}
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\hol{GEN\_TAC} & remove outermost all-quantifier \\
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\hol{DISCH\_TAC} & move antecedent of goal into assumptions \\
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\hol{CONJ\_TAC} & splits conjunctive goal \\
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\hol{STRIP\_TAC} & splits on outermost connective (combination\\
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& \quad of \hol{GEN\_TAC}, \hol{CONJ\_TAC}, \hol{DISCH\_TAC}, \ldots) \\
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\hol{DISJ1\_TAC} & selects left disjunct \\
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\hol{DISJ2\_TAC} & selects right disjunct \\
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\hol{EQ\_TAC} & reduce Boolean equality to implications \\
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\hol{ASSUME\_TAC}\ thm & add theorem to list of assumptions \\
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\hol{EXISTS\_TAC} term & provide witness for existential goal \\
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\end{tabular}
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\end{frame}
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\begin{frame}
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\frametitle{Tacticals}
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\begin{itemize}
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\item tacticals are SML functions that combine tactics to form new tactics
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\item common workflow
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\begin{itemize}
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\item develop large tactic interactively
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\item using \hol{goalStack} and editor support to execute tactics one by one
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\item combine tactics manually with tacticals to create larger tactics
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\item finally end up with one large tactic that solves your goal
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\item use \hol{prove} or \hol{store\_thm} instead of \hol{goalStack}
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\end{itemize}
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\item make sure to \alert{clearly mark proof structure} by \eg
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\begin{itemize}
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\item use indentation
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\item use parentheses
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\item use appropriate connectives
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\item \ldots
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\end{itemize}
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\item goalStack commands like \hol{e} or \hol{g} should not appear in your final proof
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Some Basic Tacticals}
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\begin{tabular}{lll}
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tac1 \hol{>>} tac2 & \hol{THEN}, \hol{\textbsl{}\textbsl{}} & applies tactics in sequence \\
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tac \hol{>|} tacL & \hol{THENL} & applies list of tactics to subgoals \\
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tac1 \hol{>-} tac2 & \hol{THEN1} & applies tac2 to the first subgoal of tac1 \\
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\hol{REPEAT} tac & \hol{rpt} & repeats tac until it fails \\
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\hol{NTAC} n tac & & apply tac n times \\
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\hol{REVERSE} tac & \hol{reverse} & reverses the order of subgoals \\
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tac1 \hol{ORELSE} tac2 & & applies tac1 only if tac2 fails \\
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\hol{TRY} tac & & do nothing if tac fails \\
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\hol{ALL\_TAC} & \hol{all\_tac} & do nothing \\
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\hol{NO\_TAC} & & fail
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\end{tabular}
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\end{frame}
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\begin{frame}
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\frametitle{Basic Rewrite Tactics}
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\begin{itemize}
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\item (equational) rewriting is at the core of HOL's automation
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\item we will discuss it in detail later
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\item details complex, but basic usage is straightforward
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\begin{itemize}
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\item given a theorem \hol{rewr\_thm} of form \hol{|- P\ x = Q\ x} and a term \hol{t}
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\item rewriting \hol{t} with \hol{rewr\_thm} means
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\item replacing each occurrence of a term \hol{P c} for some \hol{c} with \hol{Q c} in \hol{t}
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\end{itemize}
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\item \alert{warning:} rewriting may loop\\Example: rewriting with theorem \hol{|- X <=> (X \holAnd{} T)}
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\end{itemize}
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\begin{tabular}{ll}
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\hol{REWRITE\_TAC} thms & rewrite goal using equations found\\
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& in given list of theorems \\
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\hol{ASM\_REWRITE\_TAC} thms & in addition use assumptions \\
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\hol{ONCE\_REWRITE\_TAC} thms & rewrite once in goal using equations\\
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\hol{ONCE\_ASM\_REWRITE\_TAC} thms & rewrite once using assumptions
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\end{tabular}
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\end{frame}
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\begin{frame}
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\frametitle{Case-Split and Induction Tactics}
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\begin{tabular}{ll}
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\hol{Induct\_on} `term` & induct on \texttt{term} \\
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\hol{Induct} & induct on all-quantifier \\
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\hol{Cases\_on} `term` & case-split on \texttt{term} \\
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\hol{Cases} & case-split on all-quantifier \\
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\hol{MATCH\_MP\_TAC} thm & apply rule \\
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\hol{IRULE\_TAC} thm & generalised apply rule
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\end{tabular}
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\end{frame}
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\begin{frame}
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\frametitle{Assumption Tactics}
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\begin{tabular}{ll}
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\hol{POP\_ASSUM} thm-tac & use and remove first assumption \\
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& \-\quad common usage \hol{POP\_ASSUM MP\_TAC} \\[1em]
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\hol{PAT\_ASSUM} term thm-tac& use (and remove) first \\
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\-\quad also \hol{PAT\_X\_ASSUM} term thm-tac& \quad assumption matching pattern \\[1em]
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\hol{WEAKEN\_TAC} term-pred & removes first assumption \\
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& \quad{}satisfying predicate
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\end{tabular}
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\end{frame}
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\begin{frame}
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\frametitle{Decision Procedure Tactics}
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\begin{itemize}
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\item decision procedures try to solve the current goal completely
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\item they either succeed or fail
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\item no partial progress
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\item decision procedures vital for automation
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\end{itemize}
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\bigskip
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\begin{tabular}{ll}
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\hol{TAUT\_TAC} & propositional logic tautology checker \\
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\hol{DECIDE\_TAC} & linear arithmetic for \texttt{num} \\
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\hol{METIS\_TAC} thms & first order prover \\
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\hol{numLib.ARITH\_TAC} & Presburger arithmetic \\
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\hol{intLib.ARITH\_TAC} & uses Omega test
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\end{tabular}
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\end{frame}
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\begin{frame}
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\frametitle{Subgoal Tactics}
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\begin{itemize}
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\item it is vital to structure your proofs well
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\begin{itemize}
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\item improved maintainability
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\item improved readability
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\item improved reusability
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\item saves time in medium-run
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\end{itemize}
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\item therefore, use many small lemmata
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\item also, use many explicit subgoals
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\end{itemize}
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\bigskip
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\begin{tabular}{ll}
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`term-frag` \hol{by} tac & show term with tac and\\
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& add it to assumptions \\
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`term-frag` \hol{suffices\_by} tac & show it suffices to prove term
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\end{tabular}
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\end{frame}
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\begin{frame}
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\frametitle{Term Fragments / Term Quotations}
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\begin{itemize}
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\item notice that \hol{by} and \hol{suffices\_by} take \emph{term fragments}
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\item term fragments are also called \emph{term quotations}
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\item they represent (partially) unparsed terms
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\item parsing takes place during execution of tactic in context of goal
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\item this helps to avoid type annotations
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\item however, this means syntax errors show late as well
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\item the library \emph{Q} defines many tactics using term fragments
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\end{itemize}
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\end{frame}
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\section{Examples}
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\begin{frame}
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\frametitle{Importance of Exercises}
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\begin{itemize}
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\item here many tactics are presented in a very short amount of time
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\item there are many, many more important tactics out there
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\item few people can learn a programming language just by reading manuals
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\item similar few people can learn HOL just by reading and listening
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\item you should write your own proofs and play around with these tactics
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\item solving the exercises is highly recommended\\(and actually required if you want credits for this course)
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\end{itemize}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Tactical Proof - Example I - Slide 1}
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\begin{itemize}
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\item we want to prove \hol{!l.\ LENGTH (APPEND l l) = 2 * LENGTH l}
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\item first step: set up goal on \hol{goalStack}
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\item at same time start writing proof script
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\end{itemize}
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\begin{block}{Proof Script}
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\begin{semiverbatim}\small
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val LENGTH_APPEND_SAME = prove (
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``!l. LENGTH (APPEND l l) = 2 * LENGTH l``,
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\end{semiverbatim}
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\end{block}
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\begin{block}{Actions}
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\begin{itemize}
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\item run \texttt{g ``!l.\ LENGTH (APPEND l l) = 2 * LENGTH l``}
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\item this is done by hol-mode
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\item move cursor inside term and press \texttt{M-h g}\\
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(menu-entry \texttt{HOL - Goalstack - New goal})
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\end{itemize}
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\end{block}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Tactical Proof - Example I - Slide 2}
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\begin{block}{Current Goal}
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\begin{semiverbatim}\small
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!l. LENGTH (l ++ l) = 2 * LENGTH l
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\end{semiverbatim}
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\end{block}
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\begin{itemize}
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\item the outermost connective is an all-quantifier
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\item let's get rid of it via \hol{GEN\_TAC}
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\end{itemize}
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\begin{block}{Proof Script}
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\begin{semiverbatim}\small
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val LENGTH_APPEND_SAME = prove (
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``!l. LENGTH (l ++ l) = 2 * LENGTH l``,
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GEN_TAC
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\end{semiverbatim}
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\end{block}
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\begin{block}{Actions}
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\begin{itemize}
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\item run \texttt{e GEN\_TAC}
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\item this is done by hol-mode
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\item mark line with \texttt{GEN\_TAC} and press \texttt{M-h e}\\
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(menu-entry \texttt{HOL - Goalstack - Apply tactic})
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\end{itemize}
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\end{block}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Tactical Proof - Example I - Slide 3}
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\begin{block}{Current Goal}
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\begin{semiverbatim}\small
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LENGTH (l ++ l) = 2 * LENGTH l
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\end{semiverbatim}
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\end{block}
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\begin{itemize}
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\item \hol{LENGTH} of \hol{APPEND} can be simplified
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\item let's search an appropriate lemma with \ml{DB.match}
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\end{itemize}
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\begin{block}{Actions}
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\begin{itemize}
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\item run \ml{DB.print\_match [] ``LENGTH (\_ ++ \_)``}
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\item this is done via hol-mode
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\item press \texttt{M-h m} and enter term pattern\\
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(menu-entry \texttt{HOL - Misc - DB match})
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\item this finds the theorem \ml{listTheory.LENGTH\_APPEND}\\
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\hol{|- !l1 l2. LENGTH (l1 ++ l2) = LENGTH l1 + LENGTH l2}
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\end{itemize}
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\end{block}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Tactical Proof - Example I - Slide 4}
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\begin{block}{Current Goal}
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\begin{semiverbatim}\small
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LENGTH (l ++ l) = 2 * LENGTH l
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\end{semiverbatim}
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\end{block}
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\begin{itemize}
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\item let's rewrite with found theorem \ml{listTheory.LENGTH\_APPEND}
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\end{itemize}
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\begin{block}{Proof Script}
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\begin{semiverbatim}\small
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val LENGTH_APPEND_SAME = prove (
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``!l. LENGTH (APPEND l l) = 2 * LENGTH l``,
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GEN_TAC >>
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REWRITE_TAC[listTheory.LENGTH\_APPEND]
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\end{semiverbatim}
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\end{block}
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\begin{block}{Actions}
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\begin{itemize}
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\item connect the new tactic with tactical \hol{>>} (\hol{THEN})
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\item use hol-mode to expand the new tactic
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\end{itemize}
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\end{block}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Tactical Proof - Example I - Slide 5}
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\begin{block}{Current Goal}
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\begin{semiverbatim}\small
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LENGTH l + LENGTH l = 2 * LENGTH l
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\end{semiverbatim}
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\end{block}
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\begin{itemize}
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\item let's search a theorem for simplifying \hol{2 * LENGTH l}
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\item prepare for extending the previous rewrite tactic
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\end{itemize}
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\begin{block}{Proof Script}
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\begin{semiverbatim}\small
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val LENGTH_APPEND_SAME = prove (
|
||
|
|
``!l. LENGTH (APPEND l l) = 2 * LENGTH l``,
|
||
|
|
GEN_TAC >>
|
||
|
|
REWRITE_TAC[listTheory.LENGTH\_APPEND]
|
||
|
|
\end{semiverbatim}
|
||
|
|
\end{block}
|
||
|
|
|
||
|
|
\begin{block}{Actions}
|
||
|
|
\begin{itemize}
|
||
|
|
\item \hol{DB.match} finds theorem \hol{arithmeticTheory.TIMES2}
|
||
|
|
\item press \texttt{M-h b} and undo last tactic expansion\\
|
||
|
|
(menu-entry \texttt{HOL - Goalstack - Back up})
|
||
|
|
\end{itemize}
|
||
|
|
\end{block}
|
||
|
|
\end{frame}
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
\begin{frame}[fragile]
|
||
|
|
\frametitle{Tactical Proof - Example I - Slide 6}
|
||
|
|
|
||
|
|
\begin{block}{Current Goal}
|
||
|
|
\begin{semiverbatim}\small
|
||
|
|
LENGTH (l ++ l) = 2 * LENGTH l
|
||
|
|
\end{semiverbatim}
|
||
|
|
\end{block}
|
||
|
|
|
||
|
|
\begin{itemize}
|
||
|
|
\item extend the previous rewrite tactic
|
||
|
|
\item finish proof
|
||
|
|
\end{itemize}
|
||
|
|
|
||
|
|
\begin{block}{Proof Script}
|
||
|
|
\begin{semiverbatim}\small
|
||
|
|
val LENGTH_APPEND_SAME = prove (
|
||
|
|
``!l. LENGTH (APPEND l l) = 2 * LENGTH l``,
|
||
|
|
GEN_TAC >>
|
||
|
|
REWRITE_TAC[listTheory.LENGTH\_APPEND, arithmeticTheory.TIMES2]);
|
||
|
|
\end{semiverbatim}
|
||
|
|
\end{block}
|
||
|
|
|
||
|
|
\begin{block}{Actions}
|
||
|
|
\begin{itemize}
|
||
|
|
\item add \hol{TIMES2} to the list of theorems used by rewrite tactic
|
||
|
|
\item use hol-mode to expand the extended rewrite tactic
|
||
|
|
\item goal is solved, so let's add closing parenthesis and semicolon
|
||
|
|
\end{itemize}
|
||
|
|
\end{block}
|
||
|
|
\end{frame}
|
||
|
|
|
||
|
|
|
||
|
|
\begin{frame}[fragile]
|
||
|
|
\frametitle{Tactical Proof - Example I - Slide 7}
|
||
|
|
|
||
|
|
\begin{itemize}
|
||
|
|
\item we have a finished tactic proving our goal
|
||
|
|
\item notice that \hol{GEN\_TAC} is not needed
|
||
|
|
\item let's polish the proof script
|
||
|
|
\end{itemize}
|
||
|
|
|
||
|
|
\begin{block}{Proof Script}
|
||
|
|
\begin{semiverbatim}\small
|
||
|
|
val LENGTH_APPEND_SAME = prove (
|
||
|
|
``!l. LENGTH (APPEND l l) = 2 * LENGTH l``,
|
||
|
|
GEN_TAC >>
|
||
|
|
REWRITE_TAC[listTheory.LENGTH\_APPEND, arithmeticTheory.TIMES2]);
|
||
|
|
\end{semiverbatim}
|
||
|
|
\end{block}
|
||
|
|
|
||
|
|
\begin{block}{Polished Proof Script}
|
||
|
|
\begin{semiverbatim}\small
|
||
|
|
val LENGTH_APPEND_SAME = prove (
|
||
|
|
``!l. LENGTH (APPEND l l) = 2 * LENGTH l``,
|
||
|
|
REWRITE_TAC[listTheory.LENGTH\_APPEND, arithmeticTheory.TIMES2]);
|
||
|
|
\end{semiverbatim}
|
||
|
|
\end{block}
|
||
|
|
\end{frame}
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
\begin{frame}[fragile]
|
||
|
|
\frametitle{Tactical Proof - Example II - Slide 1}
|
||
|
|
|
||
|
|
\begin{itemize}
|
||
|
|
\item let's prove something slightly more complicated
|
||
|
|
\item drop old goal by pressing \texttt{M-h d}\\
|
||
|
|
(menu-entry \texttt{HOL - Goalstack - Drop goal})
|
||
|
|
\item set up goal on \hol{goalStack} (\texttt{M-h g})
|
||
|
|
\item at same time start writing proof script
|
||
|
|
\end{itemize}
|
||
|
|
|
||
|
|
\begin{block}{Proof Script}
|
||
|
|
\begin{semiverbatim}\small
|
||
|
|
val NOT_ALL_DISTINCT_LEMMA = prove (``!x1 x2 x3 l1 l2 l3.
|
||
|
|
(MEM x1 l1 \holAnd{} MEM x2 l2 \holAnd{} MEM x3 l3) \holAnd{}
|
||
|
|
((x1 <= x2) \holAnd{} (x2 <= x3) \holAnd{} x3 <= SUC x1) ==>
|
||
|
|
~(ALL_DISTINCT (l1 ++ l2 ++ l3))``,
|
||
|
|
\end{semiverbatim}
|
||
|
|
\end{block}
|
||
|
|
|
||
|
|
\end{frame}
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
\begin{frame}[fragile]
|
||
|
|
\frametitle{Tactical Proof - Example II - Slide 2}
|
||
|
|
|
||
|
|
\begin{block}{Current Goal}
|
||
|
|
\begin{semiverbatim}\small
|
||
|
|
!x1 x2 x3 l1 l2 l3.
|
||
|
|
(MEM x1 l1 \holAnd{} MEM x2 l2 \holAnd{} MEM x3 l3) \holAnd{}
|
||
|
|
x1 <= x2 \holAnd{} x2 <= x3 \holAnd{} x3 <= SUC x1 ==>
|
||
|
|
~ALL_DISTINCT (l1 ++ l2 ++ l3)
|
||
|
|
\end{semiverbatim}
|
||
|
|
\end{block}
|
||
|
|
|
||
|
|
|
||
|
|
\begin{itemize}
|
||
|
|
\item let's strip the goal
|
||
|
|
\end{itemize}
|
||
|
|
|
||
|
|
\begin{block}{Proof Script}
|
||
|
|
\begin{semiverbatim}\small
|
||
|
|
val NOT_ALL_DISTINCT_LEMMA = prove (``!x1 x2 x3 l1 l2 l3.
|
||
|
|
(MEM x1 l1 \holAnd{} MEM x2 l2 \holAnd{} MEM x3 l3) \holAnd{}
|
||
|
|
((x1 <= x2) \holAnd{} (x2 <= x3) \holAnd{} x3 <= SUC x1) ==>
|
||
|
|
~(ALL_DISTINCT (l1 ++ l2 ++ l3))``,
|
||
|
|
REPEAT STRIP\_TAC
|
||
|
|
\end{semiverbatim}
|
||
|
|
\end{block}
|
||
|
|
\end{frame}
|
||
|
|
|
||
|
|
\begin{frame}[fragile]
|
||
|
|
\frametitle{Tactical Proof - Example II - Slide 2}
|
||
|
|
|
||
|
|
\begin{block}{Current Goal}
|
||
|
|
\begin{semiverbatim}\small
|
||
|
|
!x1 x2 x3 l1 l2 l3.
|
||
|
|
(MEM x1 l1 \holAnd{} MEM x2 l2 \holAnd{} MEM x3 l3) \holAnd{}
|
||
|
|
x1 <= x2 \holAnd{} x2 <= x3 \holAnd{} x3 <= SUC x1 ==>
|
||
|
|
~ALL_DISTINCT (l1 ++ l2 ++ l3)
|
||
|
|
\end{semiverbatim}
|
||
|
|
\end{block}
|
||
|
|
|
||
|
|
\begin{itemize}
|
||
|
|
\item let's strip the goal
|
||
|
|
\end{itemize}
|
||
|
|
|
||
|
|
\begin{block}{Proof Script}
|
||
|
|
\begin{semiverbatim}\small
|
||
|
|
val LENGTH_APPEND_SAME = prove (
|
||
|
|
``!l. LENGTH (APPEND l l) = 2 * LENGTH l``,
|
||
|
|
REPEAT STRIP\_TAC
|
||
|
|
\end{semiverbatim}
|
||
|
|
\end{block}
|
||
|
|
|
||
|
|
\begin{block}{Actions}
|
||
|
|
\begin{itemize}
|
||
|
|
\item add \hol{REPEAT STRIP\_TAC} to proof script
|
||
|
|
\item expand this tactic using hol-mode
|
||
|
|
\end{itemize}
|
||
|
|
\end{block}
|
||
|
|
\end{frame}
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
\begin{frame}[fragile]
|
||
|
|
\frametitle{Tactical Proof - Example II - Slide 3}
|
||
|
|
|
||
|
|
\begin{block}{Current Goal}
|
||
|
|
\begin{semiverbatim}\small
|
||
|
|
F
|
||
|
|
------------------------------------
|
||
|
|
0. MEM x1 l1 4. x2 <= x3
|
||
|
|
1. MEM x2 l2 5. x3 <= SUC x1
|
||
|
|
2. MEM x3 l3 6. ALL_DISTINCT (l1 ++ l2 ++ l3)
|
||
|
|
3. x1 <= x2
|
||
|
|
\end{semiverbatim}
|
||
|
|
\end{block}
|
||
|
|
|
||
|
|
\begin{itemize}
|
||
|
|
\item oops, we did too much, we would like to keep \texttt{ALL\_DISTINCT} in goal
|
||
|
|
\end{itemize}
|
||
|
|
\begin{block}{Proof Script}
|
||
|
|
\begin{semiverbatim}\small
|
||
|
|
val NOT_ALL_DISTINCT_LEMMA = prove (``...``,
|
||
|
|
REPEAT GEN\_TAC >> STRIP\_TAC
|
||
|
|
\end{semiverbatim}
|
||
|
|
\end{block}
|
||
|
|
|
||
|
|
\begin{block}{Actions}
|
||
|
|
\begin{itemize}
|
||
|
|
\item undo \hol{REPEAT STRIP\_TAC} (\texttt{M-h b})
|
||
|
|
\item expand more fine-tuned strip tactic
|
||
|
|
\end{itemize}
|
||
|
|
\end{block}
|
||
|
|
\end{frame}
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
\begin{frame}[fragile]
|
||
|
|
\frametitle{Tactical Proof - Example II - Slide 4}
|
||
|
|
|
||
|
|
\begin{block}{Current Goal}
|
||
|
|
\begin{semiverbatim}\small
|
||
|
|
~ALL_DISTINCT (l1 ++ l2 ++ l3)
|
||
|
|
------------------------------------
|
||
|
|
0. MEM x1 l1 3. x1 <= x2
|
||
|
|
1. MEM x2 l2 4. x2 <= x3
|
||
|
|
2. MEM x3 l3 5. x3 <= SUC x1
|
||
|
|
\end{semiverbatim}
|
||
|
|
\end{block}
|
||
|
|
|
||
|
|
\begin{itemize}
|
||
|
|
\item now let's simplify \hol{ALL\_DISTINCT}
|
||
|
|
\item search suitable theorems with \hol{DB.match}
|
||
|
|
\item use them with rewrite tactic
|
||
|
|
\end{itemize}
|
||
|
|
\begin{block}{Proof Script}
|
||
|
|
\begin{semiverbatim}\small
|
||
|
|
val NOT_ALL_DISTINCT_LEMMA = prove (``...``,
|
||
|
|
REPEAT GEN\_TAC >> STRIP\_TAC >>
|
||
|
|
REWRITE\_TAC[listTheory.ALL_DISTINCT\_APPEND, listTheory.MEM\_APPEND]
|
||
|
|
\end{semiverbatim}
|
||
|
|
\end{block}
|
||
|
|
\end{frame}
|
||
|
|
|
||
|
|
|
||
|
|
\begin{frame}[fragile]
|
||
|
|
\frametitle{Tactical Proof - Example II - Slide 5}
|
||
|
|
|
||
|
|
\begin{block}{Current Goal}
|
||
|
|
\begin{semiverbatim}\scriptsize
|
||
|
|
~((ALL_DISTINCT l1 \holAnd{} ALL_DISTINCT l2 \holAnd{} !e. MEM e l1 ==> ~MEM e l2) \holAnd{}
|
||
|
|
ALL_DISTINCT l3 \holAnd{} !e. MEM e l1 \holOr{} MEM e l2 ==> ~MEM e l3)
|
||
|
|
------------------------------------
|
||
|
|
0. MEM x1 l1 3. x1 <= x2
|
||
|
|
1. MEM x2 l2 4. x2 <= x3
|
||
|
|
2. MEM x3 l3 5. x3 <= SUC x1
|
||
|
|
\end{semiverbatim}
|
||
|
|
\end{block}
|
||
|
|
|
||
|
|
\begin{itemize}
|
||
|
|
\item from assumptions 3, 4 and 5 we know \hol{x2 = x1 \holOr{} x2 = x3}
|
||
|
|
\item let's deduce this fact by \hol{DECIDE\_TAC}
|
||
|
|
\end{itemize}
|
||
|
|
\begin{block}{Proof Script}
|
||
|
|
\begin{semiverbatim}\scriptsize
|
||
|
|
val NOT_ALL_DISTINCT_LEMMA = prove (``...``,
|
||
|
|
REPEAT GEN\_TAC >> STRIP\_TAC >>
|
||
|
|
REWRITE\_TAC[listTheory.ALL_DISTINCT\_APPEND, listTheory.MEM\_APPEND] >>
|
||
|
|
`(x2 = x1) \holOr{} (x2 = x3)` by DECIDE_TAC
|
||
|
|
\end{semiverbatim}
|
||
|
|
\end{block}
|
||
|
|
\end{frame}
|
||
|
|
|
||
|
|
|
||
|
|
\begin{frame}[fragile]
|
||
|
|
\frametitle{Tactical Proof - Example II - Slide 6}
|
||
|
|
|
||
|
|
\begin{block}{Current Goals --- 2 subgoals, one for each disjunct}
|
||
|
|
\begin{semiverbatim}\scriptsize
|
||
|
|
~((ALL_DISTINCT l1 \holAnd{} ALL_DISTINCT l2 \holAnd{} !e. MEM e l1 ==> ~MEM e l2) \holAnd{}
|
||
|
|
ALL_DISTINCT l3 \holAnd{} !e. MEM e l1 \holOr{} MEM e l2 ==> ~MEM e l3)
|
||
|
|
------------------------------------
|
||
|
|
0. MEM x1 l1 4. x2 <= x3
|
||
|
|
1. MEM x2 l2 5. x3 <= SUC x1
|
||
|
|
2. MEM x3 l3 6a. x2 = x1
|
||
|
|
3. x1 <= x2 6b. x2 = x3
|
||
|
|
\end{semiverbatim}
|
||
|
|
\end{block}
|
||
|
|
|
||
|
|
\begin{itemize}
|
||
|
|
\item both goals are easily solved by first-order reasoning
|
||
|
|
\item let's use \hol{METIS\_TAC[]} for both subgoals
|
||
|
|
\end{itemize}
|
||
|
|
\begin{block}{Proof Script}
|
||
|
|
\begin{semiverbatim}\scriptsize
|
||
|
|
val NOT_ALL_DISTINCT_LEMMA = prove (``...``,
|
||
|
|
REPEAT GEN\_TAC >> STRIP\_TAC >>
|
||
|
|
REWRITE\_TAC[listTheory.ALL_DISTINCT\_APPEND, listTheory.MEM\_APPEND] >>
|
||
|
|
`(x2 = x1) \holOr{} (x2 = x3)` by DECIDE_TAC >> (
|
||
|
|
METIS\_TAC[]
|
||
|
|
));
|
||
|
|
\end{semiverbatim}
|
||
|
|
\end{block}
|
||
|
|
\end{frame}
|
||
|
|
|
||
|
|
|
||
|
|
\begin{frame}[fragile]
|
||
|
|
\frametitle{Tactical Proof - Example II - Slide 7}
|
||
|
|
|
||
|
|
\begin{block}{Finished Proof Script}
|
||
|
|
\begin{semiverbatim}\scriptsize
|
||
|
|
val NOT_ALL_DISTINCT_LEMMA = prove (
|
||
|
|
``!x1 x2 x3 l1 l2 l3.
|
||
|
|
(MEM x1 l1 \holAnd{} MEM x2 l2 \holAnd{} MEM x3 l3) \holAnd{}
|
||
|
|
((x1 <= x2) \holAnd{} (x2 <= x3) \holAnd{} x3 <= SUC x1) ==>
|
||
|
|
~(ALL_DISTINCT (l1 ++ l2 ++ l3))``,
|
||
|
|
REPEAT GEN\_TAC >> STRIP\_TAC >>
|
||
|
|
REWRITE\_TAC[listTheory.ALL_DISTINCT\_APPEND, listTheory.MEM\_APPEND] >>
|
||
|
|
`(x2 = x1) \holOr{} (x2 = x3)` by DECIDE_TAC >> (
|
||
|
|
METIS\_TAC[]
|
||
|
|
));
|
||
|
|
\end{semiverbatim}
|
||
|
|
\end{block}
|
||
|
|
\begin{itemize}
|
||
|
|
\item notice that proof structure is explicit
|
||
|
|
\item parentheses and indentation used to mark new subgoals
|
||
|
|
\end{itemize}
|
||
|
|
\end{frame}
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
%%% Local Variables:
|
||
|
|
%%% mode: latex
|
||
|
|
%%% TeX-master: "current"
|
||
|
|
%%% End:
|