author  wenzelm 
Mon, 24 Aug 1998 19:12:13 +0200  
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permissions  rwrr 
104  1 
%% $Id$ 
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\chapter{Substitution Tactics} \label{substitution} 

323  3 
\index{tactics!substitution(}\index{equality(} 
4 

104  5 
Replacing equals by equals is a basic form of reasoning. Isabelle supports 
332  6 
several kinds of equality reasoning. {\bf Substitution} means replacing 
104  7 
free occurrences of~$t$ by~$u$ in a subgoal. This is easily done, given an 
332  8 
equality $t=u$, provided the logic possesses the appropriate rule. The 
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tactic \texttt{hyp_subst_tac} performs substitution even in the assumptions. 
332  10 
But it works via objectlevel implication, and therefore must be specially 
11 
set up for each suitable objectlogic. 

104  12 

13 
Substitution should not be confused with objectlevel {\bf rewriting}. 

14 
Given equalities of the form $t=u$, rewriting replaces instances of~$t$ by 

15 
corresponding instances of~$u$, and continues until it reaches a normal 

16 
form. Substitution handles `oneoff' replacements by particular 

332  17 
equalities while rewriting handles general equations. 
3950  18 
Chapter~\ref{chap:simplification} discusses Isabelle's rewriting tactics. 
104  19 

20 

323  21 
\section{Substitution rules} 
22 
\index{substitution!rules}\index{*subst theorem} 

23 
Many logics include a substitution rule of the form 

3108  24 
$$ 
25 
\List{\Var{a}=\Var{b}; \Var{P}(\Var{a})} \Imp 

26 
\Var{P}(\Var{b}) \eqno(subst) 

27 
$$ 

104  28 
In backward proof, this may seem difficult to use: the conclusion 
29 
$\Var{P}(\Var{b})$ admits far too many unifiers. But, if the theorem {\tt 

30 
eqth} asserts $t=u$, then \hbox{\tt eqth RS subst} is the derived rule 

31 
\[ \Var{P}(t) \Imp \Var{P}(u). \] 

32 
Provided $u$ is not an unknown, resolution with this rule is 

33 
wellbehaved.\footnote{Unifying $\Var{P}(u)$ with a formula~$Q$ 

34 
expresses~$Q$ in terms of its dependence upon~$u$. There are still $2^k$ 

35 
unifiers, if $Q$ has $k$ occurrences of~$u$, but Isabelle ensures that 

36 
the first unifier includes all the occurrences.} To replace $u$ by~$t$ in 

37 
subgoal~$i$, use 

38 
\begin{ttbox} 

332  39 
resolve_tac [eqth RS subst] \(i\){\it.} 
104  40 
\end{ttbox} 
41 
To replace $t$ by~$u$ in 

42 
subgoal~$i$, use 

43 
\begin{ttbox} 

332  44 
resolve_tac [eqth RS ssubst] \(i\){\it,} 
104  45 
\end{ttbox} 
323  46 
where \tdxbold{ssubst} is the `swapped' substitution rule 
3108  47 
$$ 
48 
\List{\Var{a}=\Var{b}; \Var{P}(\Var{b})} \Imp 

49 
\Var{P}(\Var{a}). \eqno(ssubst) 

50 
$$ 

323  51 
If \tdx{sym} denotes the symmetry rule 
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\(\Var{a}=\Var{b}\Imp\Var{b}=\Var{a}\), then \texttt{ssubst} is just 
104  53 
\hbox{\tt sym RS subst}. Many logics with equality include the rules {\tt 
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subst} and \texttt{ssubst}, as well as \texttt{refl}, \texttt{sym} and \texttt{trans} 
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(for the usual equality laws). Examples include \texttt{FOL} and \texttt{HOL}, 
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but not \texttt{CTT} (Constructive Type Theory). 
104  57 

58 
Elimresolution is wellbehaved with assumptions of the form $t=u$. 

59 
To replace $u$ by~$t$ or $t$ by~$u$ in subgoal~$i$, use 

60 
\begin{ttbox} 

332  61 
eresolve_tac [subst] \(i\) {\rm or} eresolve_tac [ssubst] \(i\){\it.} 
104  62 
\end{ttbox} 
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3108  64 
Logics \HOL, {\FOL} and {\ZF} define the tactic \ttindexbold{stac} by 
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\begin{ttbox} 
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fun stac eqth = CHANGED o rtac (eqth RS ssubst); 
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\end{ttbox} 
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Now \texttt{stac~eqth} is like \texttt{resolve_tac [eqth RS ssubst]} but with the 
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valuable property of failing if the substitution has no effect. 
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104  71 

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\section{Substitution in the hypotheses} 

323  73 
\index{assumptions!substitution in} 
104  74 
Substitution rules, like other rules of natural deduction, do not affect 
75 
the assumptions. This can be inconvenient. Consider proving the subgoal 

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\[ \List{c=a; c=b} \Imp a=b. \] 

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Calling \texttt{eresolve_tac\ts[ssubst]\ts\(i\)} simply discards the 
104  78 
assumption~$c=a$, since $c$ does not occur in~$a=b$. Of course, we can 
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work out a solution. First apply \texttt{eresolve_tac\ts[subst]\ts\(i\)}, 
104  80 
replacing~$a$ by~$c$: 
4374  81 
\[ c=b \Imp c=b \] 
104  82 
Equality reasoning can be difficult, but this trivial proof requires 
83 
nothing more sophisticated than substitution in the assumptions. 

323  84 
Objectlogics that include the rule~$(subst)$ provide tactics for this 
104  85 
purpose: 
86 
\begin{ttbox} 

323  87 
hyp_subst_tac : int > tactic 
88 
bound_hyp_subst_tac : int > tactic 

104  89 
\end{ttbox} 
323  90 
\begin{ttdescription} 
104  91 
\item[\ttindexbold{hyp_subst_tac} {\it i}] 
323  92 
selects an equality assumption of the form $t=u$ or $u=t$, where $t$ is a 
93 
free variable or parameter. Deleting this assumption, it replaces $t$ 

94 
by~$u$ throughout subgoal~$i$, including the other assumptions. 

95 

96 
\item[\ttindexbold{bound_hyp_subst_tac} {\it i}] 

97 
is similar but only substitutes for parameters (bound variables). 

98 
Uses for this are discussed below. 

99 
\end{ttdescription} 

104  100 
The term being replaced must be a free variable or parameter. Substitution 
101 
for constants is usually unhelpful, since they may appear in other 

102 
theorems. For instance, the best way to use the assumption $0=1$ is to 

103 
contradict a theorem that states $0\not=1$, rather than to replace 0 by~1 

104 
in the subgoal! 

105 

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Substitution for unknowns, such as $\Var{x}=0$, is a bad idea: we might prove 
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the subgoal more easily by instantiating~$\Var{x}$ to~1. 
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Substitution for free variables is unhelpful if they appear in the 
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premises of a rule being derived: the substitution affects objectlevel 
104  110 
assumptions, not metalevel assumptions. For instance, replacing~$a$ 
323  111 
by~$b$ could make the premise~$P(a)$ worthless. To avoid this problem, use 
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\texttt{bound_hyp_subst_tac}; alternatively, call \ttindex{cut_facts_tac} to 
323  113 
insert the atomic premises as objectlevel assumptions. 
104  114 

115 

116 
\section{Setting up {\tt hyp_subst_tac}} 

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Many Isabelle objectlogics, such as \texttt{FOL}, \texttt{HOL} and their 
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descendants, come with \texttt{hyp_subst_tac} already defined. A few others, 
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such as \texttt{CTT}, do not support this tactic because they lack the 
104  120 
rule~$(subst)$. When defining a new logic that includes a substitution 
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rule and implication, you must set up \texttt{hyp_subst_tac} yourself. It 
104  122 
is packaged as the \ML{} functor \ttindex{HypsubstFun}, which takes the 
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argument signature~\texttt{HYPSUBST_DATA}: 
104  124 
\begin{ttbox} 
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signature HYPSUBST_DATA = 

126 
sig 

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structure Simplifier : SIMPLIFIER 
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val dest_Trueprop : term > term 
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val dest_eq : term > term*term*typ 
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val dest_imp : term > term*term 
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val eq_reflection : thm (* a=b ==> a==b *) 
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val imp_intr : thm (* (P ==> Q) ==> P>Q *) 
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val rev_mp : thm (* [ P; P>Q ] ==> Q *) 
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val subst : thm (* [ a=b; P(a) ] ==> P(b) *) 
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val sym : thm (* a=b ==> b=a *) 
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val thin_refl : thm (* [x=x; P] ==> P *) 
104  137 
end; 
138 
\end{ttbox} 

139 
Thus, the functor requires the following items: 

323  140 
\begin{ttdescription} 
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\item[Simplifier] should be an instance of the simplifier (see 
3950  142 
Chapter~\ref{chap:simplification}). 
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\item[\ttindexbold{dest_Trueprop}] should coerce a metalevel formula to the 
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corresponding objectlevel one. Typically, it should return $P$ when 
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applied to the term $\texttt{Trueprop}\,P$ (see example below). 
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\item[\ttindexbold{dest_eq}] should return the triple~$(t,u,T)$, where $T$ is 
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the type of~$t$ and~$u$, when applied to the \ML{} term that 
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represents~$t=u$. For other terms, it should raise an exception. 
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\item[\ttindexbold{dest_imp}] should return the pair~$(P,Q)$ when applied to 
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the \ML{} term that represents the implication $P\imp Q$. For other terms, 
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it should raise an exception. 
104  155 

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\item[\tdxbold{eq_reflection}] is the theorem discussed 
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in~\S\ref{sec:settingupsimp}. 
104  158 

323  159 
\item[\tdxbold{imp_intr}] should be the implies introduction 
104  160 
rule $(\Var{P}\Imp\Var{Q})\Imp \Var{P}\imp\Var{Q}$. 
161 

323  162 
\item[\tdxbold{rev_mp}] should be the `reversed' implies elimination 
104  163 
rule $\List{\Var{P}; \;\Var{P}\imp\Var{Q}} \Imp \Var{Q}$. 
164 

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\item[\tdxbold{subst}] should be the substitution rule 
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$\List{\Var{a}=\Var{b};\; \Var{P}(\Var{a})} \Imp \Var{P}(\Var{b})$. 
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\item[\tdxbold{sym}] should be the symmetry rule 
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$\Var{a}=\Var{b}\Imp\Var{b}=\Var{a}$. 
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\item[\tdxbold{thin_refl}] should be the rule 
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$\List{\Var{a}=\Var{a};\; \Var{P}} \Imp \Var{P}$, which is used to erase 
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trivial equalities. 
323  174 
\end{ttdescription} 
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% 
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The functor resides in file \texttt{Provers/hypsubst.ML} in the Isabelle 
104  177 
distribution directory. It is not sensitive to the precise formalization 
178 
of the objectlogic. It is not concerned with the names of the equality 

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and implication symbols, or the types of formula and terms. 
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Coding the functions \texttt{dest_Trueprop}, \texttt{dest_eq} and 
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\texttt{dest_imp} requires knowledge of Isabelle's representation of terms. 
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For \texttt{FOL}, they are declared by 
104  184 
\begin{ttbox} 
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fun dest_Trueprop (Const ("Trueprop", _) $ P) = P 
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 dest_Trueprop t = raise TERM ("dest_Trueprop", [t]); 
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fun dest_eq (Const("op =",T) $ t $ u) = (t, u, domain_type T) 
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fun dest_imp (Const("op >",_) $ A $ B) = (A, B) 
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 dest_imp t = raise TERM ("dest_imp", [t]); 
104  192 
\end{ttbox} 
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Recall that \texttt{Trueprop} is the coercion from type~$o$ to type~$prop$, 
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while \hbox{\tt op =} is the internal name of the infix operator~\texttt{=}. 
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Function \ttindexbold{domain_type}, given the function type $S\To T$, returns 
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the type~$S$. Patternmatching expresses the function concisely, using 
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wildcards~({\tt_}) for the types. 
104  198 

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The tactic \texttt{hyp_subst_tac} works as follows. First, it identifies a 
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suitable equality assumption, possibly reorienting it using~\texttt{sym}. 
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Then it moves other assumptions into the conclusion of the goal, by repeatedly 
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calling \texttt{etac~rev_mp}. Then, it uses \texttt{asm_full_simp_tac} or 
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\texttt{ssubst} to substitute throughout the subgoal. (If the equality 
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involves unknowns then it must use \texttt{ssubst}.) Then, it deletes the 
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equality. Finally, it moves the assumptions back to their original positions 
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by calling \hbox{\tt resolve_tac\ts[imp_intr]}. 
104  207 

323  208 
\index{equality)}\index{tactics!substitution)} 
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