author  nipkow 
Wed, 04 Aug 2004 11:25:08 +0200  
changeset 15106  e8cef6993701 
parent 11181  d04f57b91166 
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104  1 
%%%THIS DOCUMENTS THE OBSOLETE SIMPLIFIER!!!! 
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\chapter{Simplification} \label{simpchap} 

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\index{simplification(} 

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Objectlevel rewriting is not primitive in Isabelle. For efficiency, 

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perhaps it ought to be. On the other hand, it is difficult to conceive of 

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a general mechanism that could accommodate the diversity of rewriting found 

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in different logics. Hence rewriting in Isabelle works via resolution, 

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using unknowns as placeholders for simplified terms. This chapter 

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describes a generic simplification package, the functor~\ttindex{SimpFun}, 

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which expects the basic laws of equational logic and returns a suite of 

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simplification tactics. The code lives in 

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\verb$Provers/simp.ML$. 

13 

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This rewriting package is not as general as one might hope (using it for {\tt 

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HOL} is not quite as convenient as it could be; rewriting modulo equations is 

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not supported~\ldots) but works well for many logics. It performs 

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conditional and unconditional rewriting and handles multiple reduction 

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relations and local assumptions. It also has a facility for automatic case 

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splits by expanding conditionals like {\it ifthenelse\/} during rewriting. 

20 

21 
For many of Isabelle's logics ({\tt FOL}, {\tt ZF}, {\tt LCF} and {\tt HOL}) 

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the simplifier has been set up already. Hence we start by describing the 

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functions provided by the simplifier  those functions exported by 

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\ttindex{SimpFun} through its result signature \ttindex{SIMP} shown in 

286  25 
Fig.\ts\ref{SIMP}. 
104  26 

27 

28 
\section{Simplification sets} 

29 
\index{simplification sets} 

30 
The simplification tactics are controlled by {\bf simpsets}, which consist of 

31 
three things: 

32 
\begin{enumerate} 

33 
\item {\bf Rewrite rules}, which are theorems like 

34 
$\Var{m} + succ(\Var{n}) = succ(\Var{m} + \Var{n})$. {\bf Conditional} 

35 
rewrites such as $m<n \Imp m/n = 0$ are permitted. 

36 
\index{rewrite rules} 

37 

38 
\item {\bf Congruence rules}, which typically have the form 

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\index{congruence rules} 

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\[ \List{\Var{x@1} = \Var{y@1}; \ldots; \Var{x@n} = \Var{y@n}} \Imp 

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f(\Var{x@1},\ldots,\Var{x@n}) = f(\Var{y@1},\ldots,\Var{y@n}). 

42 
\] 

43 

44 
\item The {\bf autotactic}, which attempts to solve the simplified 

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subgoal, say by recognizing it as a tautology. 

46 
\end{enumerate} 

47 

48 
\subsection{Congruence rules} 

49 
Congruence rules enable the rewriter to simplify subterms. Without a 

50 
congruence rule for the function~$g$, no argument of~$g$ can be rewritten. 

51 
Congruence rules can be generalized in the following ways: 

52 

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{\bf Additional assumptions} are allowed: 

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\[ \List{\Var{P@1} \bimp \Var{Q@1};\; \Var{Q@1} \Imp \Var{P@2} \bimp \Var{Q@2}} 

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\Imp (\Var{P@1} \imp \Var{P@2}) \bimp (\Var{Q@1} \imp \Var{Q@2}) 

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\] 

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This rule assumes $Q@1$, and any rewrite rules it contains, while 

1100  58 
simplifying~$P@2$. Such `local' assumptions are effective for rewriting 
104  59 
formulae such as $x=0\imp y+x=y$. 
60 

61 
{\bf Additional quantifiers} are allowed, typically for binding operators: 

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\[ \List{\Forall z. \Var{P}(z) \bimp \Var{Q}(z)} \Imp 

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\forall x.\Var{P}(x) \bimp \forall x.\Var{Q}(x) 

64 
\] 

65 

66 
{\bf Different equalities} can be mixed. The following example 

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enables the transition from formula rewriting to term rewriting: 

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\[ \List{\Var{x@1}=\Var{y@1};\Var{x@2}=\Var{y@2}} \Imp 

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(\Var{x@1}=\Var{x@2}) \bimp (\Var{y@1}=\Var{y@2}) 

70 
\] 

71 
\begin{warn} 

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It is not necessary to assert a separate congruence rule for each constant, 

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provided your logic contains suitable substitution rules. The function {\tt 

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mk_congs} derives congruence rules from substitution 

75 
rules~\S\ref{simptactics}. 

76 
\end{warn} 

77 

78 

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\begin{figure} 

80 
\indexbold{*SIMP} 

81 
\begin{ttbox} 

82 
infix 4 addrews addcongs delrews delcongs setauto; 

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signature SIMP = 

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sig 

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type simpset 

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val empty_ss : simpset 

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val addcongs : simpset * thm list > simpset 

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val addrews : simpset * thm list > simpset 

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val delcongs : simpset * thm list > simpset 

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val delrews : simpset * thm list > simpset 

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val print_ss : simpset > unit 

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val setauto : simpset * (int > tactic) > simpset 

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val ASM_SIMP_CASE_TAC : simpset > int > tactic 

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val ASM_SIMP_TAC : simpset > int > tactic 

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val CASE_TAC : simpset > int > tactic 

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val SIMP_CASE2_TAC : simpset > int > tactic 

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val SIMP_THM : simpset > thm > thm 

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val SIMP_TAC : simpset > int > tactic 

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val SIMP_CASE_TAC : simpset > int > tactic 

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val mk_congs : theory > string list > thm list 

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val mk_typed_congs : theory > (string*string) list > thm list 

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val tracing : bool ref 

103 
end; 

104 
\end{ttbox} 

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\caption{The signature {\tt SIMP}} \label{SIMP} 

106 
\end{figure} 

107 

108 

109 
\subsection{The abstract type {\tt simpset}}\label{simpsimpsets} 

110 
Simpsets are values of the abstract type \ttindexbold{simpset}. They are 

111 
manipulated by the following functions: 

112 
\index{simplification setsbold} 

323  113 
\begin{ttdescription} 
104  114 
\item[\ttindexbold{empty_ss}] 
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is the empty simpset. It has no congruence or rewrite rules and its 

116 
autotactic always fails. 

117 

323  118 
\item[$ss$ \ttindexbold{addcongs} $thms$] 
104  119 
is the simpset~$ss$ plus the congruence rules~$thms$. 
120 

323  121 
\item[$ss$ \ttindexbold{delcongs} $thms$] 
104  122 
is the simpset~$ss$ minus the congruence rules~$thms$. 
123 

323  124 
\item[$ss$ \ttindexbold{addrews} $thms$] 
104  125 
is the simpset~$ss$ plus the rewrite rules~$thms$. 
126 

323  127 
\item[$ss$ \ttindexbold{delrews} $thms$] 
104  128 
is the simpset~$ss$ minus the rewrite rules~$thms$. 
129 

323  130 
\item[$ss$ \ttindexbold{setauto} $tacf$] 
104  131 
is the simpset~$ss$ with $tacf$ for its autotactic. 
132 

133 
\item[\ttindexbold{print_ss} $ss$] 

134 
prints all the congruence and rewrite rules in the simpset~$ss$. 

323  135 
\end{ttdescription} 
104  136 
Adding a rule to a simpset already containing it, or deleting one 
137 
from a simpset not containing it, generates a warning message. 

138 

139 
In principle, any theorem can be used as a rewrite rule. Before adding a 

140 
theorem to a simpset, {\tt addrews} preprocesses the theorem to extract the 

141 
maximum amount of rewriting from it. Thus it need not have the form $s=t$. 

142 
In {\tt FOL} for example, an atomic formula $P$ is transformed into the 

143 
rewrite rule $P \bimp True$. This preprocessing is not fixed but logic 

144 
dependent. The existing logics like {\tt FOL} are fairly clever in this 

145 
respect. For a more precise description see {\tt mk_rew_rules} in 

146 
\S\ref{SimpFuninput}. 

147 

148 
The autotactic is applied after simplification to solve a goal. This may 

149 
be the overall goal or some subgoal that arose during conditional 

150 
rewriting. Calling ${\tt auto_tac}~i$ must either solve exactly 

151 
subgoal~$i$ or fail. If it succeeds without reducing the number of 

152 
subgoals by one, havoc and strange exceptions may result. 

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A typical autotactic is {\tt ares_tac [TrueI]}, which attempts proof by 

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assumption and resolution with the theorem $True$. In explicitly typed 

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logics, the autotactic can be used to solve simple type checking 

156 
obligations. Some applications demand a sophisticated autotactic such as 

157 
{\tt fast_tac}, but this could make simplification slow. 

158 

159 
\begin{warn} 

160 
Rewriting never instantiates unknowns in subgoals. (It uses 

161 
\ttindex{match_tac} rather than \ttindex{resolve_tac}.) However, the 

162 
autotactic is permitted to instantiate unknowns. 

163 
\end{warn} 

164 

165 

166 
\section{The simplification tactics} \label{simptactics} 

167 
\index{simplification!tacticsbold} 

168 
\index{tactics!simplificationbold} 

169 
The actual simplification work is performed by the following tactics. The 

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rewriting strategy is strictly bottom up. Conditions in conditional rewrite 

171 
rules are solved recursively before the rewrite rule is applied. 

172 

173 
There are two basic simplification tactics: 

323  174 
\begin{ttdescription} 
104  175 
\item[\ttindexbold{SIMP_TAC} $ss$ $i$] 
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simplifies subgoal~$i$ using the rules in~$ss$. It may solve the 

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subgoal completely if it has become trivial, using the autotactic 

178 
(\S\ref{simpsimpsets}). 

179 

180 
\item[\ttindexbold{ASM_SIMP_TAC}] 

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is like \verb$SIMP_TAC$, but also uses assumptions as additional 

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rewrite rules. 

323  183 
\end{ttdescription} 
104  184 
Many logics have conditional operators like {\it ifthenelse}. If the 
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simplifier has been set up with such case splits (see~\ttindex{case_splits} 

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in \S\ref{SimpFuninput}), there are tactics which automatically alternate 

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between simplification and case splitting: 

323  188 
\begin{ttdescription} 
104  189 
\item[\ttindexbold{SIMP_CASE_TAC}] 
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is like {\tt SIMP_TAC} but also performs automatic case splits. 

191 
More precisely, after each simplification phase the tactic tries to apply a 

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theorem in \ttindex{case_splits}. If this succeeds, the tactic calls 

193 
itself recursively on the result. 

194 

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\item[\ttindexbold{ASM_SIMP_CASE_TAC}] 

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is like {\tt SIMP_CASE_TAC}, but also uses assumptions for 

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rewriting. 

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\item[\ttindexbold{SIMP_CASE2_TAC}] 

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is like {\tt SIMP_CASE_TAC}, but also tries to solve the 

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preconditions of conditional simplification rules by repeated case splits. 

202 

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\item[\ttindexbold{CASE_TAC}] 

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tries to break up a goal using a rule in 

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\ttindex{case_splits}. 

206 

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\item[\ttindexbold{SIMP_THM}] 

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simplifies a theorem using assumptions and case splitting. 

323  209 
\end{ttdescription} 
104  210 
Finally there are two useful functions for generating congruence 
211 
rules for constants and free variables: 

323  212 
\begin{ttdescription} 
104  213 
\item[\ttindexbold{mk_congs} $thy$ $cs$] 
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computes a list of congruence rules, one for each constant in $cs$. 

215 
Remember that the name of an infix constant 

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\verb$+$ is \verb$op +$. 

217 

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\item[\ttindexbold{mk_typed_congs}] 

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computes congruence rules for explicitly typed free variables and 

220 
constants. Its second argument is a list of name and type pairs. Names 

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can be either free variables like {\tt P}, or constants like \verb$op =$. 

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For example in {\tt FOL}, the pair 

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\verb$("f","'a => 'a")$ generates the rule \verb$?x = ?y ==> f(?x) = f(?y)$. 

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Such congruence rules are necessary for goals with free variables whose 

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arguments need to be rewritten. 

323  226 
\end{ttdescription} 
104  227 
Both functions work correctly only if {\tt SimpFun} has been supplied with the 
228 
necessary substitution rules. The details are discussed in 

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\S\ref{SimpFuninput} under {\tt subst_thms}. 

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\begin{warn} 

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Using the simplifier effectively may take a bit of experimentation. In 

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particular it may often happen that simplification stops short of what you 

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expected or runs forever. To diagnose these problems, the simplifier can be 

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traced. The reference variable \ttindexbold{tracing} controls the output of 

235 
tracing information. 

236 
\index{tracing!of simplification} 

237 
\end{warn} 

238 

239 

240 
\section{Example: using the simplifier} 

241 
\index{simplification!example} 

242 
Assume we are working within {\tt FOL} and that 

323  243 
\begin{ttdescription} 
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\item[Nat.thy] is a theory including the constants $0$, $Suc$ and $+$, 

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\item[add_0] is the rewrite rule $0+n = n$, 

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\item[add_Suc] is the rewrite rule $Suc(m)+n = Suc(m+n)$, 

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\item[induct] is the induction rule 

104  248 
$\List{P(0); \Forall x. P(x)\Imp P(Suc(x))} \Imp P(n)$. 
323  249 
\item[FOL_ss] is a basic simpset for {\tt FOL}. 
250 
\end{ttdescription} 

104  251 
We generate congruence rules for $Suc$ and for the infix operator~$+$: 
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\begin{ttbox} 

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val nat_congs = mk_congs Nat.thy ["Suc", "op +"]; 

254 
prths nat_congs; 

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{\out ?Xa = ?Ya ==> Suc(?Xa) = Suc(?Ya)} 

256 
{\out [ ?Xa = ?Ya; ?Xb = ?Yb ] ==> ?Xa + ?Xb = ?Ya + ?Yb} 

257 
\end{ttbox} 

258 
We create a simpset for natural numbers by extending~{\tt FOL_ss}: 

259 
\begin{ttbox} 

260 
val add_ss = FOL_ss addcongs nat_congs 

261 
addrews [add_0, add_Suc]; 

262 
\end{ttbox} 

263 
Proofs by induction typically involve simplification:\footnote 

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{These examples reside on the file {\tt FOL/ex/nat.ML}.} 

265 
\begin{ttbox} 

266 
goal Nat.thy "m+0 = m"; 

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{\out Level 0} 

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{\out m + 0 = m} 

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{\out 1. m + 0 = m} 

270 
\ttbreak 

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by (res_inst_tac [("n","m")] induct 1); 

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{\out Level 1} 

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{\out m + 0 = m} 

274 
{\out 1. 0 + 0 = 0} 

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{\out 2. !!x. x + 0 = x ==> Suc(x) + 0 = Suc(x)} 

276 
\end{ttbox} 

277 
Simplification solves the first subgoal: 

278 
\begin{ttbox} 

279 
by (SIMP_TAC add_ss 1); 

280 
{\out Level 2} 

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{\out m + 0 = m} 

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{\out 1. !!x. x + 0 = x ==> Suc(x) + 0 = Suc(x)} 

283 
\end{ttbox} 

284 
The remaining subgoal requires \ttindex{ASM_SIMP_TAC} in order to use the 

285 
induction hypothesis as a rewrite rule: 

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\begin{ttbox} 

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by (ASM_SIMP_TAC add_ss 1); 

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{\out Level 3} 

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{\out m + 0 = m} 

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{\out No subgoals!} 

291 
\end{ttbox} 

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The next proof is similar. 

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\begin{ttbox} 

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goal Nat.thy "m+Suc(n) = Suc(m+n)"; 

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{\out Level 0} 

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{\out m + Suc(n) = Suc(m + n)} 

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{\out 1. m + Suc(n) = Suc(m + n)} 

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\ttbreak 

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by (res_inst_tac [("n","m")] induct 1); 

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{\out Level 1} 

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{\out m + Suc(n) = Suc(m + n)} 

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{\out 1. 0 + Suc(n) = Suc(0 + n)} 

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{\out 2. !!x. x + Suc(n) = Suc(x + n) ==> Suc(x) + Suc(n) = Suc(Suc(x) + n)} 

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\end{ttbox} 

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Using the tactical \ttindex{ALLGOALS}, a single command simplifies all the 

306 
subgoals: 

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\begin{ttbox} 

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by (ALLGOALS (ASM_SIMP_TAC add_ss)); 

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{\out Level 2} 

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{\out m + Suc(n) = Suc(m + n)} 

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{\out No subgoals!} 

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\end{ttbox} 

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Some goals contain free function variables. The simplifier must have 

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congruence rules for those function variables, or it will be unable to 

315 
simplify their arguments: 

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\begin{ttbox} 

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val f_congs = mk_typed_congs Nat.thy [("f","nat => nat")]; 

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val f_ss = add_ss addcongs f_congs; 

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prths f_congs; 

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{\out ?Xa = ?Ya ==> f(?Xa) = f(?Ya)} 

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\end{ttbox} 

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Here is a conjecture to be proved for an arbitrary function~$f$ satisfying 

323 
the law $f(Suc(n)) = Suc(f(n))$: 

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\begin{ttbox} 

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val [prem] = goal Nat.thy 

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"(!!n. f(Suc(n)) = Suc(f(n))) ==> f(i+j) = i+f(j)"; 

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{\out Level 0} 

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{\out f(i + j) = i + f(j)} 

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{\out 1. f(i + j) = i + f(j)} 

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\ttbreak 

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by (res_inst_tac [("n","i")] induct 1); 

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{\out Level 1} 

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{\out f(i + j) = i + f(j)} 

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{\out 1. f(0 + j) = 0 + f(j)} 

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{\out 2. !!x. f(x + j) = x + f(j) ==> f(Suc(x) + j) = Suc(x) + f(j)} 

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\end{ttbox} 

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We simplify each subgoal in turn. The first one is trivial: 

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\begin{ttbox} 

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by (SIMP_TAC f_ss 1); 

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{\out Level 2} 

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{\out f(i + j) = i + f(j)} 

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{\out 1. !!x. f(x + j) = x + f(j) ==> f(Suc(x) + j) = Suc(x) + f(j)} 

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\end{ttbox} 

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The remaining subgoal requires rewriting by the premise, shown 

345 
below, so we add it to {\tt f_ss}: 

346 
\begin{ttbox} 

347 
prth prem; 

348 
{\out f(Suc(?n)) = Suc(f(?n)) [!!n. f(Suc(n)) = Suc(f(n))]} 

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by (ASM_SIMP_TAC (f_ss addrews [prem]) 1); 

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{\out Level 3} 

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{\out f(i + j) = i + f(j)} 

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{\out No subgoals!} 

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\end{ttbox} 

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355 

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\section{Setting up the simplifier} \label{SimpFuninput} 

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\index{simplification!setting upbold} 

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To set up a simplifier for a new logic, the \ML\ functor 

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\ttindex{SimpFun} needs to be supplied with theorems to justify 

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rewriting. A rewrite relation must be reflexive and transitive; symmetry 

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is not necessary. Hence the package is also applicable to nonsymmetric 

362 
relations such as occur in operational semantics. In the sequel, $\gg$ 

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denotes some {\bf reduction relation}: a binary relation to be used for 

364 
rewriting. Several reduction relations can be used at once. In {\tt FOL}, 

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both $=$ (on terms) and $\bimp$ (on formulae) can be used for rewriting. 

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367 
The argument to {\tt SimpFun} is a structure with signature 

368 
\ttindexbold{SIMP_DATA}: 

369 
\begin{ttbox} 

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signature SIMP_DATA = 

371 
sig 

372 
val case_splits : (thm * string) list 

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val dest_red : term > term * term * term 

374 
val mk_rew_rules : thm > thm list 

375 
val norm_thms : (thm*thm) list 

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val red1 : thm 

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val red2 : thm 

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val refl_thms : thm list 

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val subst_thms : thm list 

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val trans_thms : thm list 

381 
end; 

382 
\end{ttbox} 

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The components of {\tt SIMP_DATA} need to be instantiated as follows. Many 

384 
of these components are lists, and can be empty. 

323  385 
\begin{ttdescription} 
104  386 
\item[\ttindexbold{refl_thms}] 
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supplies reflexivity theorems of the form $\Var{x} \gg 

388 
\Var{x}$. They must not have additional premises as, for example, 

389 
$\Var{x}\in\Var{A} \Imp \Var{x} = \Var{x}\in\Var{A}$ in type theory. 

390 

391 
\item[\ttindexbold{trans_thms}] 

392 
supplies transitivity theorems of the form 

393 
$\List{\Var{x}\gg\Var{y}; \Var{y}\gg\Var{z}} \Imp {\Var{x}\gg\Var{z}}$. 

394 

395 
\item[\ttindexbold{red1}] 

396 
is a theorem of the form $\List{\Var{P}\gg\Var{Q}; 

397 
\Var{P}} \Imp \Var{Q}$, where $\gg$ is a relation between formulae, such as 

398 
$\bimp$ in {\tt FOL}. 

399 

400 
\item[\ttindexbold{red2}] 

401 
is a theorem of the form $\List{\Var{P}\gg\Var{Q}; 

402 
\Var{Q}} \Imp \Var{P}$, where $\gg$ is a relation between formulae, such as 

403 
$\bimp$ in {\tt FOL}. 

404 

405 
\item[\ttindexbold{mk_rew_rules}] 

406 
is a function that extracts rewrite rules from theorems. A rewrite rule is 

407 
a theorem of the form $\List{\ldots}\Imp s \gg t$. In its simplest form, 

408 
{\tt mk_rew_rules} maps a theorem $t$ to the singleton list $[t]$. More 

409 
sophisticated versions may do things like 

410 
\[ 

411 
\begin{array}{l@{}r@{\quad\mapsto\quad}l} 

412 
\mbox{create formula rewrites:}& P & [P \bimp True] \\[.5ex] 

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\mbox{remove negations:}& \neg P & [P \bimp False] \\[.5ex] 
104  414 
\mbox{create conditional rewrites:}& P\imp s\gg t & [P\Imp s\gg t] \\[.5ex] 
415 
\mbox{break up conjunctions:}& 

416 
(s@1 \gg@1 t@1) \conj (s@2 \gg@2 t@2) & [s@1 \gg@1 t@1, s@2 \gg@2 t@2] 

417 
\end{array} 

418 
\] 

419 
The more theorems are turned into rewrite rules, the better. The function 

420 
is used in two places: 

421 
\begin{itemize} 

422 
\item 

423 
$ss$~\ttindex{addrews}~$thms$ applies {\tt mk_rew_rules} to each element of 

424 
$thms$ before adding it to $ss$. 

425 
\item 

426 
simplification with assumptions (as in \ttindex{ASM_SIMP_TAC}) uses 

427 
{\tt mk_rew_rules} to turn assumptions into rewrite rules. 

428 
\end{itemize} 

429 

430 
\item[\ttindexbold{case_splits}] 

431 
supplies expansion rules for case splits. The simplifier is designed 

432 
for rules roughly of the kind 

433 
\[ \Var{P}(if(\Var{Q},\Var{x},\Var{y})) \bimp (\Var{Q} \imp \Var{P}(\Var{x})) 

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\conj (\neg\Var{Q} \imp \Var{P}(\Var{y})) 
104  435 
\] 
436 
but is insensitive to the form of the righthand side. Other examples 

437 
include product types, where $split :: 

438 
(\alpha\To\beta\To\gamma)\To\alpha*\beta\To\gamma$: 

439 
\[ \Var{P}(split(\Var{f},\Var{p})) \bimp (\forall a~b. \Var{p} = 

440 
{<}a,b{>} \imp \Var{P}(\Var{f}(a,b))) 

441 
\] 

442 
Each theorem in the list is paired with the name of the constant being 

443 
eliminated, {\tt"if"} and {\tt"split"} in the examples above. 

444 

445 
\item[\ttindexbold{norm_thms}] 

446 
supports an optimization. It should be a list of pairs of rules of the 

447 
form $\Var{x} \gg norm(\Var{x})$ and $norm(\Var{x}) \gg \Var{x}$. These 

448 
introduce and eliminate {\tt norm}, an arbitrary function that should be 

449 
used nowhere else. This function serves to tag subterms that are in normal 

450 
form. Such rules can speed up rewriting significantly! 

451 

452 
\item[\ttindexbold{subst_thms}] 

453 
supplies substitution rules of the form 

454 
\[ \List{\Var{x} \gg \Var{y}; \Var{P}(\Var{x})} \Imp \Var{P}(\Var{y}) \] 

455 
They are used to derive congruence rules via \ttindex{mk_congs} and 

456 
\ttindex{mk_typed_congs}. If $f :: [\tau@1,\cdots,\tau@n]\To\tau$ is a 

457 
constant or free variable, the computation of a congruence rule 

458 
\[\List{\Var{x@1} \gg@1 \Var{y@1}; \ldots; \Var{x@n} \gg@n \Var{y@n}} 

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\Imp f(\Var{x@1},\ldots,\Var{x@n}) \gg f(\Var{y@1},\ldots,\Var{y@n}) \] 

460 
requires a reflexivity theorem for some reduction ${\gg} :: 

461 
\alpha\To\alpha\To\sigma$ such that $\tau$ is an instance of $\alpha$. If a 

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suitable reflexivity law is missing, no congruence rule for $f$ can be 

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generated. Otherwise an $n$ary congruence rule of the form shown above is 

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derived, subject to the availability of suitable substitution laws for each 

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argument position. 

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A substitution law is suitable for argument $i$ if it 

468 
uses a reduction ${\gg@i} :: \alpha@i\To\alpha@i\To\sigma@i$ such that 

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$\tau@i$ is an instance of $\alpha@i$. If a suitable substitution law for 

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argument $i$ is missing, the $i^{th}$ premise of the above congruence rule 

471 
cannot be generated and hence argument $i$ cannot be rewritten. In the 

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worst case, if there are no suitable substitution laws at all, the derived 

473 
congruence simply degenerates into a reflexivity law. 

474 

475 
\item[\ttindexbold{dest_red}] 

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takes reductions apart. Given a term $t$ representing the judgement 

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\mbox{$a \gg b$}, \verb$dest_red$~$t$ should return a triple $(c,ta,tb)$ 

478 
where $ta$ and $tb$ represent $a$ and $b$, and $c$ is a term of the form 

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\verb$Const(_,_)$, the reduction constant $\gg$. 

480 

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Suppose the logic has a coercion function like $Trueprop::o\To prop$, as do 

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{\tt FOL} and~{\tt HOL}\@. If $\gg$ is a binary operator (not necessarily 

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infix), the following definition does the job: 

484 
\begin{verbatim} 

485 
fun dest_red( _ $ (c $ ta $ tb) ) = (c,ta,tb); 

486 
\end{verbatim} 

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The wildcard pattern {\tt_} matches the coercion function. 

323  488 
\end{ttdescription} 
104  489 

490 

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\section{A sample instantiation} 

9695  492 
Here is the instantiation of {\tt SIMP_DATA} for FOL. The code for {\tt 
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mk_rew_rules} is not shown; see the file {\tt FOL/simpdata.ML}. 

104  494 
\begin{ttbox} 
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structure FOL_SimpData : SIMP_DATA = 

496 
struct 

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val refl_thms = [ \(\Var{x}=\Var{x}\), \(\Var{P}\bimp\Var{P}\) ] 

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val trans_thms = [ \(\List{\Var{x}=\Var{y};\Var{y}=\Var{z}}\Imp\Var{x}=\Var{z}\), 

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\(\List{\Var{P}\bimp\Var{Q};\Var{Q}\bimp\Var{R}}\Imp\Var{P}\bimp\Var{R}\) ] 

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val red1 = \(\List{\Var{P}\bimp\Var{Q}; \Var{P}} \Imp \Var{Q}\) 

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val red2 = \(\List{\Var{P}\bimp\Var{Q}; \Var{Q}} \Imp \Var{P}\) 

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val mk_rew_rules = ... 

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val case_splits = [ \(\Var{P}(if(\Var{Q},\Var{x},\Var{y})) \bimp\) 

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d04f57b91166
renamed addaltern to addafter, addSaltern to addSafter
oheimb
parents:
9695
diff
changeset

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\((\Var{Q} \imp \Var{P}(\Var{x})) \conj (\neg\Var{Q} \imp \Var{P}(\Var{y}))\) ] 
104  505 
val norm_thms = [ (\(\Var{x}=norm(\Var{x})\),\(norm(\Var{x})=\Var{x}\)), 
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(\(\Var{P}\bimp{}NORM(\Var{P}\)), \(NORM(\Var{P})\bimp\Var{P}\)) ] 

507 
val subst_thms = [ \(\List{\Var{x}=\Var{y}; \Var{P}(\Var{x})}\Imp\Var{P}(\Var{y})\) ] 

508 
val dest_red = fn (_ $ (opn $ lhs $ rhs)) => (opn,lhs,rhs) 

509 
end; 

510 
\end{ttbox} 

511 

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\index{simplification)} 